axiomatics for fuzzy rough sets abstract a fuzzy t rough set consists of a set x and a t similarity relation r on x where t is a lower semi continuous triangular norm we generalize the farinas prade definition for the upper approximation operator a i x i x of a fuzzy t rough set x r given originally for the special case t min to the case of arbitrary t we propose a new definition for the lower approximation operator a i x i x of x r our definition satisfies the two important identities a a a and a a a as well as a number of other interesting properties we provide axiomatics to fully characterize those upper and lower approximations

efficient calculation of the green function for acoustic propagation above a homogeneous impedance plane this paper is concerned with the problem of propagation from a monofrequency coherent line source above a plane of homogeneous surface impedance the solution of this problem occurs in the kernel of certain boundary integral equation formulations of acoustic propagation above an impedance boundary and the discussion of the paper is motivated by this application the paper starts by deriving representations as laplace type integrals of the solution and its first partial derivatives the evaluation of these integral representations by gauss laguerre quadrature is discussed and theoretical bounds on the truncation error are obtained specific approximations are proposed which are shown to be accurate except in the very near field for all angles of incidence and a wide range of values of surface impedance the paper finishes with derivations of partial results and analogous laplace type integral representations for the case of a point source

efficient parallel boolean matrix based algorithms for computing composite rough set approximations in information systems there may exist multiple different types of attributes like categorical attributes numerical attributes set valued attributes interval valued attributes missing attributes etc such information systems are called as composite information systems to process such attributes with rough set theory composite rough set model and corresponding matrix methods were introduced in our previous study calculation of rough set approximations of a concept is the key step for rule acquisition and attribute reduction in rough set based methods to accelerate the computation process of rough set approximations this paper first presents the boolean matrix representation of the lower and upper approximations in the composite information system then designs a parallel method for computing approximations based on matrix and implements it on multi gpu the experiments on data sets from uci and user defined data sets show that the proposed method can accelerate the computation process efficiently the multi gpu implementation achieves up to a speedup of 334 9 over the cpu implementation

efficient calculation of two dimensional periodic and waveguide acoustic green s functions new representations and efficient calculation methods are derived for the problem of propagation from an infinite regularly spaced array of coherent line sources above a homogeneous impedance plane and for the green s function for sound propagation in the canyon formed by two infinitely high parallel rigid or sound soft walls and an impedance ground surface the infinite sum of source contributions is replaced by a finite sum and the remainder is expressed as a laplace type integral a pole subtraction technique is used to remove poles in the integrand which lie near the path of integration obtaining a smooth integrand more suitable for numerical integration and a specific numerical integration method is proposed numerical experiments show highly accurate results across the frequency spectrum for a range of ground surface types it is expected that the methods proposed will prove useful in boundary element modeling of noise propagation in canyon streets and in ducts and for problems of scattering by periodic surfaces

combining numerical and linguistic information in group decision making abstract people give information about their personal preferences in many different ways depending on their background this paper deals with group decision making problems in which the solution depends on information of a different nature i e assuming that the experts express their preferences with numerical or linguistic values the aim of this paper is to present a proposal for this problem we introduce a fusion operator for numerical and linguistic information this operator combines linguistic values assessed in the same label set with numerical ones assessed in the interval 0 1 it is based on two transformation methods between numerical and linguistic values which are defined using the concept of the characteristic values proposed in this paper its application to group decision making problems is illustrated by means of a particular fusion operator guided by fuzzy majority considering that the experts express their opinions by means of fuzzy or linguistic preference relations this operator is used to develop a choice process for the alternatives allowing solutions to be obtained in line with the majority of the experts opinions

composite rough sets for dynamic data mining as a soft computing tool rough set theory has become a popular mathematical framework for pattern recognition data mining and knowledge discovery it can only deal with attributes of a specific type in the information system by using a specific binary relation however there may be attributes of multiple different types in information systems in real life applications such information systems are called as composite information systems in this paper a composite relation is proposed to process attributes of multiple different types simultaneously in composite information systems then an extended rough set model called as composite rough sets is presented we also redefine lower and upper approximations positive boundary and negative regions in composite rough sets through introducing the concepts of the relation matrix the decision matrix and the basic matrix we propose matrix based methods for computing the approximations positive boundary and negative regions in composite information systems which is crucial for feature selection and knowledge discovery moreover combined with the incremental learning technique a novel matrix based method for fast updating approximations is proposed in dynamic composite information systems extensive experiments on different data sets from uci and user defined data sets show that the proposed incremental method can process large data sets efficiently

min max attribute object bireducts on unifying models of reducts in rough set theory abstract a decision table describes a finite set of objects ob by using a finite set of condition attributes c and a finite set of decision attributes d pawlak defines attribute reducts by considering the entire decision table as a generalization we introduce the notion of min max attribute object bireducts of a sub table restricted by a pair b x of a subset of condition attributes b and a subset of objects x a pair r z is a min max attribute object bireduct of b x if and only if r is a minimal subset of b such that r and b make the same correct decisions for objects in x and z is a maximal subset of x for which b can make the correct decisions we propose the notion of the decidability of objects and introduce the decidable region of a set of objects as a generalization of the positive region of the set we define and interpret a min max attribute object bireduct based on the decidable region min max attribute object bireducts offer a general model and existing models of attribute reducts are special cases the results lead to a unified framework for studying four types of attribute reducts

scattering by infinite one dimensional rough surfaces we consider the dirichlet boundary value problem for the helmholtz equation in a non locally perturbed half plane this problem models time harmonic electromagnetic scattering by a one dimensional infinite rough perfectly conducting surface the same problem arises in acoustic scattering by a sound soft surface chandler wilde and zhang have suggested a radiation condition for this problem a generalization of the rayleigh expansion condition for diffraction gratings and uniqueness of solution has been established recently an integral equation formulation of the problem has also been proposed and in the special case when the whole boundary is both lyapunov and a small perturbation of a flat boundary the unique solvability of this integral equation has been shown by chandler wilde and ross by operator perturbation arguments in this paper we study the general case with no limit on surface amplitudes or slopes and show that the same integral equation has exactly one solution in the space of bounded and continuous functions for all wavenumbers as an important corollary we prove that for a variety of incident fields including the incident plane wave the dirichlet boundary value problem for the scattered field has a unique solution

feature selection with partition differentiation entropy for large scale data sets feature selection especially for large data sets is a challenging problem in areas such as pattern recognition machine learning and data mining with the development of data collection and storage technologies the data has become bigger than ever thus making it difficult for learning from large data sets with traditional methods in this paper we introduce the partition differentiation entropy from the viewpoint of partition in rough sets to measure the significance and uncertainty of attributes and present a feature selection method for large scale data sets based on the information theoretical measurement of attribute significance given a large scale decision information system the proposed method first divides it into small sub information systems according to the decision classes then by computing partition differentiation entropy in the sub systems the partition differentiation entropy of the attribute subset in the original decision information system is obtained accordingly the important features are selected based on the value of partition differentiation entropy the experimental results show that the idea of the proposed method is feasible and valid

active sample selection based incremental algorithm for attribute reduction with rough sets attribute reduction with rough sets is an effective technique for obtaining a compact and informative attribute set from a given dataset however traditional algorithms have no explicit provision for handling dynamic datasets where data present themselves in successive samples incremental algorithms for attribute reduction with rough sets have been recently introduced to handle dynamic datasets with large samples though they have high complexity in time and space to address the time space complexity issue of the algorithms this paper presents a novel incremental algorithm for attribute reduction with rough sets based on the adoption of an active sample selection process and an insight into the attribute reduction process this algorithm first decides whether each incoming sample is useful with respect to the current dataset by the active sample selection process a useless sample is discarded while a useful sample is selected to update a reduct at the arrival of a useful sample the attribute reduction process is then employed to guide how to add and or delete attributes in the current reduct the two processes thus constitute the theoretical framework of our algorithm the proposed algorithm is finally experimentally shown to be efficient in time and space

a fast feature weighting algorithm of data gravitation classification the data gravitation classification dgc model is a new classification method that has gained great research interest in recent years feature weights are the key parameters in dgc models because the classification performances of these models are very sensitive to such feature weights the available dgc models use wrapper like algorithms to obtain their optimised feature weights although such algorithms produce high classification accuracies but they contribute to the high computational complexities of the dgc models in this study we propose a fast feature weight algorithm for dgc models called ffw dgc we use the concepts of feature discrimination and redundancy to measure the importance of a feature after which two fuzzy subsets are constructed to respectively represent these concepts next we combine the two fuzzy subsets to compute the feature weights used in gravitational computing we conduct our experiments on 25 standard data sets and 22 imbalance data sets and compare ffw dgc with 11 kinds of classifiers including the swarm intelligence based dgc pso dgc model competitive results of ffw dgc demonstrate that it can obtain high classification accuracies but also hundreds of times of speedup ratios compared with pso dgc

a rough set approach to attribute generalization in data mining abstract this paper presents a method for updating approximations of a concept incrementally the results can be used to implement a quasi incremental algorithm for learning classification rules from very large data bases generalized by dynamic conceptual hierarchies provided by users in general the process of attribute generalization may introduce inconsistency into a generalized relation this issue is resolved by using the inductive learning algorithm lers based on rough set theory

on the generalization of fuzzy rough sets rough sets and fuzzy sets have been proved to be powerful mathematical tools to deal with uncertainty it soon raises a natural question of whether it is possible to connect rough sets and fuzzy sets the existing generalizations of fuzzy rough sets are all based on special fuzzy relations fuzzy similarity relations t similarity relations it is advantageous to generalize the fuzzy rough sets by means of arbitrary fuzzy relations and present a general framework for the study of fuzzy rough sets by using both constructive and axiomatic approaches in this paper from the viewpoint of constructive approach we first propose some definitions of upper and lower approximation operators of fuzzy sets by means of arbitrary fuzzy relations and study the relations among them the connections between special fuzzy relations and upper and lower approximation operators of fuzzy sets are also examined in axiomatic approach we characterize different classes of generalized upper and lower approximation operators of fuzzy sets by different sets of axioms the lattice and topological structures of fuzzy rough sets are also proposed in order to demonstrate that our proposed generalization of fuzzy rough sets have wider range of applications than the existing fuzzy rough sets a special lower approximation operator is applied to a fuzzy reasoning system which coincides with the mamdani algorithm

a fuzzy rough set approach for incremental feature selection on hybrid information systems in real applications there may exist many kinds of data e g boolean categorical real valued and set valued data and missing data in an information system which is called as a hybrid information system his a new hybrid distance hd in his is developed based on the value difference metric and a novel fuzzy rough set is constructed by combining the hd distance and the gaussian kernel considering the information systems often vary with time the updating mechanisms for attribute reduction feature selection are analyzed with the variation of the attribute set fuzzy rough set approaches for incremental feature selection on his are presented then two corresponding incremental algorithms are proposed respectively finally extensive experiments on eight datasets from uci and an artificial dataset show that the incremental approaches significantly outperform non incremental approaches with feature selection in the computational time

worst case complexity bounds on algorithms for computing the canonical structure of infinite abelian groups and solving systems of linear diophantine equations an o s 5 m s 2 elementary operations algorithm for computing the canonical structure of infinite abelian groups represented by a matrix of size s is presented also given is an algorithm for solving systems of linear diophantine equations say ax b in o s 3 376 log sm s 2 s 2 m s elementary operations where s is the size of a and s is the size of b the upper bounds mentioned above improve the results given by chou and collins in siam j comput 11 1982 pp 687 708

incremental updating of rough approximations in interval valued information systems under attribute generalization interval valued information system ivis is a generalized model of single valued information system in which the attribute values of objects are all interval values instead of single values the attribute set in ivis is not static but rather dynamically changing over time with the collection of new information which results in the continuous updating of rough approximations for rough set based data analysis in this paper on the basis of the similarity based rough set model in ivis we develop incremental approaches for updating rough approximations in ivis under attribute generalization which refers to the dynamic changing of attributes firstly increment relationships between the original rough approximations and the updated ones when adding or deleting an attribute set are analyzed respectively and the incremental mechanisms for updating rough approximations in ivis are introduced which carry out the computation using the previous results from the original data set along with new results then the corresponding incremental algorithms are designed based on the proposed mechanisms finally comparative experiments on data sets from uci as well as artificial data sets are conducted respectively experimental results show that the proposed incremental algorithms can effectively reduce the running time for the computation of rough approximations in comparison with the static algorithm

factoring polynomials with rational coefficients in this paper we present a polynomial time algorithm to solve the following problem given a non zero polynomial fe q x in one variable with rational coefficients find the decomposition of f into irreducible factors in q x it is well known that this is equivalent to factoring primitive polynomials fez x into irreducible factors in z x here we call f z x primitive if the greatest common divisor of its coefficients the content of f is 1 our algorithm performs well in practice cf 8 its running time measured in bit operations is o nl2 n9 log fd3

recognizing badly presented z modules finitely generated z modules have canonical decompositions when such modules are given in a finitely presented form there is a classical algorithm for computing a canonical decomposition this is the algorithm for computing the smith normal form of an integer matrix we discuss algorithms for smith normal form computation and present practical algorithms which give excellent performance for modules arising from badly presented abelian groups we investigate such issues as congruential techniques sparsity considerations pivoting strategies for gauss jordan elimination lattice basis reduction and computational complexity our results which are primarily empirical show dramatically improved performance on previous methods

pade approximants for the acoustical characteristics of rigid frame porous media in this paper it is shown that a number of theoretical models of the acoustical properties of rigid frame porous media especially those involving ratios of bessel functions of complex argument can be accurately approximated and greatly simplified by the use of pade approximation techniques in the case of the model of attenborough j acoust soc am 81 93 102 1987 rational approximations are produced for the characteristic impedance propagation constant dynamic compressibility and dynamic density as a function of frequency and the material parameters the model proposed by stinson and champoux j acoust soc am 91 685 695 1992 is approximated via rational approximation for the viscosity correction function

dynamic maintenance of approximations in set valued ordered decision systems under the attribute generalization set valued information systems are important type of data tables and generalized models of single valued information systems approximations are the focal point of approaches to knowledge discovery based on rough set theory which can be used to extract and represent the hidden knowledge in the form of decision rules attribute generalization refers to dynamic change of the attribute set in an information system with respect to the requirements of real life applications in this paper we focus on maintaining approximations dynamically in set valued ordered decision systems under the attribute generalization firstly a matrix based approach for computing approximations of upward and downward unions of decision classes is constructed by introducing the dominant and dominated matrices with respect to the dominance relation then incremental approaches for updating approximations are proposed which involves several modifications to relevant matrices without having to retrain from the start on all accumulated training data finally comparative experiments on data sets from uci as well as synthetic data sets show the proposed incremental updating methods are efficient and effective for dynamic attribute generalization

hermite normal form computation using modulo determinant arithmetic this paper describes a new class of hermite normal form solution procedures which perform modulo determinant arithmetic throughout the computation this class of procedures is shown to possess a polynomial time complexity bound which is a function of the length of the input string computational results are also given

local reduction of decision system with fuzzy rough sets one important and valuable topic in fuzzy rough sets is attribute reduction of decision system the existing attribute reductions with fuzzy rough sets consider all decision classes together and cannot identify key conditional attributes explicitly for special decision class in this paper we introduce the concept of local reduction with fuzzy rough sets for decision system the local reduction can identify key conditional attribute and offer a minimal description for every single decision class approach of discernibility matrix is employed to investigate the structure of local reduction at last several experiments are performed to show that the idea of local reduction is feasible and valid

fuzzy rough approximations for set valued data rough set theory is one of important tools of soft computing and rough approximations are the essential elements in rough set models however the existing fuzzy rough set model for set valued data which is directly constructed based on a kind of similarity relation fail to explicitly define fuzzy rough approximations to solve this issue in this paper we propose two types of fuzzy rough approximations and define two corresponding relative positive region reducts furthermore two discernibility matrices and two discernibility functions are introduced to acquire these new proposed reducts and the relationships among the new reducts and the existing reducts are also be provided theoretical analyses demonstrate that the new types of reducts have less redundancy and are more diverse no lower number of reducts than those obtained by means of the existing matrices and experimental results illustrate the new reducts found by our methods outperform those obtained by existing method

hybrid attribute reduction based on a novel fuzzy rough model and information granulation feature subset selection has become an important challenge in areas of pattern recognition machine learning and data mining as different semantics are hidden in numerical and categorical features there are two strategies for selecting hybrid attributes discretizing numerical variables or numericalize categorical features in this paper we introduce a simple and efficient hybrid attribute reduction algorithm based on a generalized fuzzy rough model a theoretic framework of fuzzy rough model based on fuzzy relations is presented which underlies a foundation for algorithm construction we derive several attribute significance measures based on the proposed fuzzy rough model and construct a forward greedy algorithm for hybrid attribute reduction the experiments show that the technique of variable precision fuzzy inclusion in computing decision positive region can get the optimal classification performance number of the selected features is the least but accuracy is the best

cohomology of congruence subgroups of sl4 z iii in two previous papers we computed cohomology groups h 5 0 n c for a range of levels n where 0 n is the congruence subgroup of sl 4 consisting of all matrices with bottom row congruent to 0 0 0 mod n in this note we update this earlier work by carrying it out for prime levels up to n 211 this requires new methods in sparse matrix reduction which are the main focus of the paper our computations involve matrices with up to 20 million nonzero entries we also make two conjectures concerning the contributions to h 5 0 n for n prime coming from eisenstein series and siegel modular forms

fuzzy rough attribute reduction with application to web categorization abstract due to the explosive growth of electronically stored information automatic methods must be developed to aid users in maintaining and using this abundance of information effectively in particular the sheer volume of redundancy present must be dealt with leaving only the information rich data to be processed this paper presents a novel approach based on an integrated use of fuzzy and rough set theories to greatly reduce this data redundancy formal concepts of fuzzy rough attribute reduction are introduced and illustrated with a simple example the work is applied to the problem of web categorization considerably reducing dimensionality with minimal loss of information experimental results show that fuzzy rough reduction is more powerful than the conventional rough set based approach classifiers that use a lower dimensional set of attributes which are retained by fuzzy rough reduction outperform those that employ more attributes returned by the existing crisp rough reduction method

fuzzy rough set based incremental attribute reduction from dynamic data with sample arriving abstract attribute reduction with fuzzy rough set is an effective technique for selecting most informative attributes from a given real valued dataset however existing algorithms for attribute reduction with fuzzy rough set have to re compute a reduct from dynamic data with sample arriving where one sample or multiple samples arrive successively this is clearly uneconomical from a computational point of view in order to efficiently find a reduct from such datasets this paper studies incremental attribute reduction with fuzzy rough sets at the arrival of one sample or multiple samples the relative discernibility relation is updated for each attribute on the basis of the updated relation an insight into the incremental process of attribute reduction with fuzzy rough sets is gained to reveal how to add new attributes into the current reduct and delete existing attributes from the current reduct applying the incremental process two incremental algorithms for attribute reduction with fuzzy rough sets are presented for one incoming sample and multiple incoming samples respectively experimental comparisons with several non incremental algorithms and the proposed incremental algorithm for one incoming sample show that our proposed incremental algorithm for multiple incoming samples can efficiently find one reduct with a comparable classification accuracy

induction of multiple fuzzy decision trees based on rough set technique the integration of fuzzy sets and rough sets can lead to a hybrid soft computing technique which has been applied successfully to many fields such as machine learning pattern recognition and image processing the key to this soft computing technique is how to set up and make use of the fuzzy attribute reduct in fuzzy rough set theory given a fuzzy information system we may find many fuzzy attribute reducts and each of them can have different contributions to decision making if only one of the fuzzy attribute reducts which may be the most important one is selected to induce decision rules some useful information hidden in the other reducts for the decision making will be losing unavoidably to sufficiently make use of the information provided by every individual fuzzy attribute reduct in a fuzzy information system this paper presents a novel induction of multiple fuzzy decision trees based on rough set technique the induction consists of three stages first several fuzzy attribute reducts are found by a similarity based approach and then a fuzzy decision tree for each fuzzy attribute reduct is generated according to the fuzzy id3 algorithm the fuzzy integral is finally considered as a fusion tool to integrate the generated decision trees which combines together all outputs of the multiple fuzzy decision trees and forms the final decision result an illustration is given to show the proposed fusion scheme a numerical experiment on real data indicates that the proposed multiple tree induction is superior to the single tree induction based on the individual reduct or on the entire feature set for learning problems with many attributes

a graph approach for fuzzy rough feature selection abstract rough sets especially fuzzy rough sets have proven to be a powerful tool for dealing with vagueness and uncertainty in data analysis fuzzy rough feature selection has been shown to be highly useful in data dimensionality reduction however many fuzzy rough feature selection algorithms are still time consuming when dealing with the large scale data sets in this paper the problem of feature selection in fuzzy rough sets is studied in the framework of graph theory we propose a new mechanism for fuzzy rough feature selection it is shown that finding the attribute reduction of a fuzzy decision system can be translated into finding the transversal of a derivative hypergraph based on the graph representation model a novel graph theoretic algorithm for fuzzy rough feature selection is proposed the performance of the proposed method is compared with those of the state of the art methods on various classification tasks experimental results show that the proposed technique outperforms all other known feature selection methods in terms of the computation time especially for the large scale data sets it demonstrates promising performance moreover our proposed method can achieve better classification accuracies with the usage of small number of features

a uniformly valid far field asymptotic expansion of the green function for two dimensional propagation above a homogeneous impedance plane abstract a generalized asymptotic expansion in the far field for the problem of cylindrical wave reflection at a homogeneous impedance plane is derived the expansion is shown to be uniformly valid over all angles of incidence and values of surface impedance including the limiting cases of zero and infinite impedance the technique used is a rigorous application of the modified steepest descent method of ott 1

feature selection in mixed data feature selection in the data with different types of feature values i e the heterogeneous or mixed data is especially of practical importance because such types of data sets widely exist in real world the key issue for feature selection in mixed data is how to properly deal with different types of the features or attributes in the data set motivated by the fuzzy rough set theory which allows different fuzzy relations to be defined for different types of attributes to measure the similarity between objects and in view of the effectiveness of entropy to measure information uncertainty we propose in this paper a fuzzy rough set based information entropy for feature selection in a mixed data set it is proved that the newly defined entropy meets the common requirement of monotonicity and can equivalently characterize the existing attribute reductions in the fuzzy rough set theory then a feature selection algorithm is formulated based on the proposed entropy and a filter wrapper method is suggested to select the best feature subset in terms of classification accuracy an extensive numerical experiment is further conducted to assess the performance of the feature selection method and the results are satisfactory highlightsa novel fuzzy rough set based information entropy is constructed for mixed data the proposed entropy can equivalently characterize the existing attribute reductions in the fuzzy rough set theory a feature selection algorithm is formulated based on the proposed entropy a filter wrapper method is suggested to select a best feature subset

learning fuzzy rules from fuzzy samples based on rough set technique although the traditional rough set theory has been a powerful mathematical tool for modeling incompleteness and vagueness its performance in dealing with initial fuzzy data is usually poor this paper makes an attempt to improve its performance by extending the traditional rough set approach to the fuzzy environment the extension is twofold one is knowledge representation and the other is knowledge reduction first we provide new definitions of fuzzy lower and upper approximations by considering the similarity between the two objects second we extend a number of underlying concepts of knowledge reduction such as the reduct and core to the fuzzy environment and use these extensions to propose a heuristic algorithm to learn fuzzy rules from initial fuzzy data finally we provide some numerical experiments to demonstrate the feasibility of the proposed algorithm one of the main contributions of this paper is that the fundamental relationship between the reducts and core of rough sets is still pertinent after the proposed extension

worst case complexity bounds on algorithms for computing the canonical structure of finite abelian groups and the hermite and smith normal forms of an integer matrix an o s 5 m s 2 algorithm for computing the canonical structure of a finite abelian group represented by an integer matrix of size s this is the smith normal form of the matrix is presented moreover an o s 3 m s 2 algorithm for computing the hermite normal form of an integer matrix of size s is given the upper bounds derived on the computational complexity of the algorithms above improve the upper bounds given by kannan and bachem in siam j comput 8 1979 pp 499 507 and chou and collins in siam j comput 11 1982 pp 687 708

incremental perspective for feature selection based on fuzzy rough sets feature selection based on fuzzy rough sets is an effective approach to select a compact feature subset that optimally predicts a given decision label despite being studied extensively most existing methods of fuzzy rough set based feature selection are restricted to computing the whole dataset in batch which is often costly or even intractable for large datasets to improve the time efficiency we investigate the incremental perspective for fuzzy rough set based feature selection assuming data can be presented in sample subsets one after another the key challenge for the incremental perspective is how to add and delete features with the subsequent arrival of sample subsets we tackle this challenge with strategies of adding and deleting features based on the relative discernibility relations that are updated as subsets arrive sequentially two incremental algorithms for fuzzy rough set based feature selection are designed based on the strategies one updates the selected features as each sample subset arrives and outputs the final feature subset where no sample subset is left the other updates the relative discernibility relations but only performs feature selection where there is no further subset arriving experimental comparisons suggest our incremental algorithms expedite fuzzy rough set based feature selection without compromising performance

asymptotically fast triangularization of matrices over rings this paper considers the problems of triangularizing and diagonalizing matrices over rings with particular emphasis on the integral case it begins with a description of fast algorithms for the computation of hermite and smith normal forms of integer matrices then it shows how to apply fast matrix multiplication techniques to the problem of triangularizing a matrix over a ring using elementary column operations these general results lead to an algorithm for triangularizing integer matrices that has a faster running time than the known hermite normal form algorithms the triangular matrix that is computed has small entries like the hermite normal form and will suffice for many applications

on the t transitivity of kernels abstract in this paper we present a view of kernels from a fuzzy set theoretical perspective indeed it turns out that kernels which are positive definite functions have to fulfill a consistency property given by the so called t transitivity of a fuzzy t equivalence relation with respect to the triangular norm t as a result we introduce a triangular norm t cos which is characterized as being the greatest one for which all kernels mapping to the unit interval and having constant 1 in its diagonal are t cos equivalences finally some applied relevant aspects are discussed as a starting point for further research