on quick attribute reduction in decision theoretic rough set models compared with the pawlak rough set model decision theoretic rough set dtrs models have relatively powerful data processing capability for complex and low quality data however the non monotonicity of the criteria used in dtrs models causes many problems particularly the inefficiency of attribute reduction algorithms which greatly narrows their applications few systematic studies have been reported on how to construct efficient reduction methods for dtrs models this paper discusses this problem and constructs an efficient reduction algorithm for dtrs models incorporating three aspects taking advantage of common characteristics of data sets and employing division techniques a fast division approach to computing equivalence classes is proposed which is the most frequently used and the most time consuming basic operation in the reduction algorithm extracting a monotonic ingredient from decision systems an effective heuristic function is constructed for the reduction algorithm which can guide searches for the algorithm and has better ability to find shorter super reducts a novel algorithm is proposed to search the power set space of a given super reduct which has a shorter average search length than the existing backtracking methods finally the reduction algorithm is proposed that combines these improvements experimental results demonstrate that these improvements are effective and the proposed reduction algorithm is relatively efficient

spin structures and spectra of ℤ 2 k manifolds we give necessary and sufficient conditions for the existence of pin and spin structures on riemannian manifolds with holonomy group 2 k for any n 4 resp n 6 we give examples of pairs of compact manifolds resp compact orientable manifolds m 1 m 2 non homeomorphic to each other that are laplace isospectral on functions and on p forms for any p and such that m 1 admits a pin resp spin structure whereas m 2 does not

tolerance spaces origins theoretical aspects and applications this article considers the origins theoretical aspects and applications of tolerance spaces in terms of the origin of tolerance spaces this article calls attention to the seminal work by j h poincare 1854 1912 and e c zeeman 1925 on establishing the foundations for tolerance spaces during the period from 1895 to 1912 poincare introduced sets of sensations and sequences of almost the same sensations as a means of characterizing the physical spectrum the perception of physical objects that are almost the same leads to a tolerance space view of visual perception as well as other forms of perception such as touch and sound roughly 60years later in 1962 zeeman formally introduced the notion of a tolerance space as a useful means of investigating a geometry of visual perception in addition to the general theory of tolerance spaces this article also carries forward earlier work on perceptual tolerance relations and considers the resemblance nearness between tolerance spaces from an information systems point of view it can be observed that tolerance spaces have proved to be fruitful in a number of research areas evidence of the utility of tolerance spaces in information systems can be seen in the introduction of tolerance rough sets tolerance approximation spaces and tolerance near sets the contribution of this article is an overview of tolerance spaces considered in the context of visual perception and a presentation of a formal basis for the study of perceptual tolerance spaces

consistency based search in feature selection feature selection is an effective technique in dealing with dimensionality reduction for classification it is used to find an optimal subset of relevant features such that the overall accuracy of classification is increased while the data size is reduced and the comprehensibility is improved feature selection methods contain two important aspects evaluation of a candidate feature subset and search through the feature space existing algorithms adopt various measures to evaluate the goodness of feature subsets this work focuses on inconsistency measure according to which a feature subset is inconsistent if there exist at least two instances with same feature values but with different class labels we compare inconsistency measure with other measures and study different search strategies such as exhaustive complete heuristic and random search that can be applied to this measure we conduct an empirical study to examine the pros and cons of these search methods give some guidelines on choosing a search method and compare the classifier error rates before and after feature selection

non solvable contractions of semisimple lie algebras in low dimension the problem of non solvable contractions of lie algebras is analysed by means of a stability theorem the problem is shown to be deeply related to the embeddings among semisimple lie algebras and the resulting branching rules for representations with this procedure we determine all deformations of indecomposable lie algebras having a nontrivial levi decomposition onto semisimple lie algebras of dimension n 8 and obtain the non solvable contractions of the latter class of algebras

a comparative study of fuzzy sets and rough sets this paper reviews and compares theories of fuzzy sets and rough sets two approaches for the formulation of fuzzy sets are reviewed one is based on many valued logic and the other is based on modal logic two views of rough sets are presented set oriented view and operator oriented view rough sets under set oriented view are closely related to fuzzy sets which leads to non truth functional fuzzy set operators both of them may be considered as deviations of classical set algebra in contrast rough sets under operator oriented view are different from fuzzy sets and may be regarded as an extension of classical set algebra

fuzzy probabilistic approximation spaces and their information measures rough set theory has proven to be an efficient tool for modeling and reasoning with uncertainty information by introducing probability into fuzzy approximation space a theory about fuzzy probabilistic approximation spaces is proposed in this paper which combines three types of uncertainty probability fuzziness and roughness into a rough set model we introduce shannon s entropy to measure information quantity implied in a pawlak s approximation space and then present a novel representation of shannon s entropy with a relation matrix based on the modified formulas some generalizations of the entropy are proposed to calculate the information in a fuzzy approximation space and a fuzzy probabilistic approximation space respectively as a result uniform representations of approximation spaces and their information measures are formed with this work

a novel attribute reduction approach for multi label data based on rough set theory multi label classification is an active research field in machine learning because of the high dimensionality of multi label data attribute reduction also known as feature selection is often necessary to improve multi label classification performance rough set theory has been widely used for attribute reduction with much success however little work has been done on applying rough set theory to attribute reduction in multi label classification in this paper a novel attribute reduction method based on rough set theory is proposed for multi label data first the uncertainties conveyed by labels are analyzed and a new type of attribute reduct is introduced called complementary decision reduct the relationships between complementary decision reduct and two representative types of attribute reducts are also investigated showing significant advantages of complementary decision reduct in revealing the uncertainties implied in multi label data second a discernibility matrix based approach is introduced for computing all complementary decision reducts and a heuristic algorithm is proposed for effectively computing a single complementary decision reduct experiments on real life data demonstrate that the proposed approach can effectively reduce unnecessary attributes and improve multi label classification accuracy

quick attribute reduct algorithm for neighborhood rough set model abstract in this paper we propose an efficient quick attribute reduct algorithm based on neighborhood rough set model in this algorithm we divide the objects records of the whole data set into a series of buckets based on their euclidean distances and then iterate each record by the sequence of buckets to calculate the positive region of neighborhood rough set model we also prove that each record s neighborhood elements can only be contained in its own bucket and its adjacent buckets thus it can reduce the iterations greatly based on the division of buckets we then present a new fast algorithm to calculate the positive region of neighborhood rough set model which can achieve a complexity of o m u m is the number of attributes u is the number of records containing in the data set furthermore with the new fast positive region computation algorithm we present a quick reduct algorithm for neighborhood rough set model and our algorithm can achieve a complexity of o m 2 u at last the efficiency of this quick reduct algorithm is proved by comparable experiments and especially this algorithm is more suitable for the reduction of big data

kloosterman sums for clifford algebras and a lower bound for the positive eigenvalues of the laplacian for congruence subgroups acting on hyperbolic spaces

probability distributions and coherent states of and algebras a new approach to probability theory based on quantum mechanical and lie algebraic ideas is proposed and developed the underlying fact is the observation that the coherent states of the heisenberg weyl su 2 and algebras in certain symmetric bosonic representations give the probability amplitudes or the square roots of the well known poisson binomial multinomial negative binomial and negative multinomial distributions in probability theory new probability distributions are derived based on coherent states of the classical algebras and in symmetric representations these new probability distributions are simple generalization of the multinomial distributions with some added new features reflecting the quantum and lie algebraic construction as byproducts simple proofs and interpretation of addition theorems of hermite polynomials are obtained from the coordinate representation of the negative multinomial states in other words these addition theorems are higher rank counterparts of the well known generating function of hermite polynomials which is essentially the coordinate representation of the ordinary heisenberg weyl coherent state

convolution singular integrals on lipschitz surfaces we prove the lp boundedness of convolution singular integral op erators on a lipschitz surface g x eo x e r 1 x e rn where g is a lipschitz function which satisfies ilvgli we have embedded r i in the clifford algebra r n with identity eo and are considering convolution with right monogenic functions 0 which satisfy 10 x i 0 so x xo x e r lxol where p a w provided there exists an l function q satisfying 0o r 0 r xern then the related convolution singular integral operator t u x lim o f yei x y n y u y dsy 0 en x u x e o jl is bounded on lp for 1 tronics macquarie university nsw 2109 australia e mail address chun li chun macadam mpce mq edu au e mail address alan mcintosh alan macadam mpce mq edu au stephen semmes department of mathematics rice university houston texas 77251 e mail address semmes rice edu this content downloaded from 157 55 39 143 on wed 15 jun 2016 04 40 33 utc all use subject to http about jstor org terms

compactification of a class of conformally flat 4 manifold in this paper we generalize huber s result on complete surfaces of finite total curvature for complete locally conformally flat 4 manifolds of positive scalar curvature with q curvature integrable where q is a variant of the chern gauss bonnet integrand we first derive the cohn vossen inequality we then establish finiteness of the topology this allows us to provide conformal compactification of such manifolds

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

neighborhood rough set based heterogeneous feature subset selection feature subset selection is viewed as an important preprocessing step for pattern recognition machine learning and data mining most of researches are focused on dealing with homogeneous feature selection namely numerical or categorical features in this paper we introduce a neighborhood rough set model to deal with the problem of heterogeneous feature subset selection as the classical rough set model can just be used to evaluate categorical features we generalize this model with neighborhood relations and introduce a neighborhood rough set model the proposed model will degrade to the classical one if we specify the size of neighborhood zero the neighborhood model is used to reduce numerical and categorical features by assigning different thresholds for different kinds of attributes in this model the sizes of the neighborhood lower and upper approximations of decisions reflect the discriminating capability of feature subsets the size of lower approximation is computed as the dependency between decision and condition attributes we use the neighborhood dependency to evaluate the significance of a subset of heterogeneous features and construct forward feature subset selection algorithms the proposed algorithms are compared with some classical techniques experimental results show that the neighborhood model based method is more flexible to deal with heterogeneous data

quick attribute reduction with generalized indiscernibility models we present a generalized indiscernibility reduction model girm and a concept of the granular structure in girm we prove that girm is compatible with three typical reduction models we present a generalized attribute reduction algorithm and a generalized positive region computing algorithm based on girm we present acceleration policies on two generalized algorithms and fast positive region computing approaches for three typical reduction models the efficiency of attribute reduction is one of the important challenges being faced in the field of big data processing although many quick attribute reduction algorithms have been proposed they are tightly coupled with their corresponding indiscernibility relations and it is difficult to extend specific acceleration policies to other reduction models in this paper we propose a generalized indiscernibility reduction model girm and a concept of the granular structure in girm which is a quantitative measurement induced from multiple indiscernibility relations and which can be used to represent the computation cost of varied models then we prove that our girm is compatible with three typical reduction models based on the proposed girm we present a generalized attribute reduction algorithm and a generalized positive region computing algorithm we perform a quantitative analysis of the computation complexities of two algorithms using the granular structure for the generalized attribute reduction we present systematic acceleration policies that can reduce the computational domain and optimize the computation of the positive region based on the granular structure we propose acceleration policies for the computation of the generalized positive region and we also propose fast positive region computation approaches for three typical reduction models experimental results for various datasets prove the efficiency of our acceleration policies in those three typical reduction models

the fourth type of covering based rough sets as a technique for granular computing rough sets deal with the vagueness and granularity in information systems covering based rough sets have been proposed to generalize this theory for wider application three types of covering based rough sets have been studied for different situations to make the theory more complete this paper proposes a fourth type of covering based rough sets compared with the existing ones the new type shows its special characteristic in the interdependency between its lower and upper approximations we carry out a systematical study of this new theory first we discuss basic properties such as normality contraction and monotone then we investigate the conditions for this type of covering based rough sets to satisfy the properties of pawlak s rough sets and study the interdependency between the lower and upper approximation operations in addition axiomatic systems for the lower and upper approximation operations are established lastly we address the relationships between this type of covering based rough sets and the three existing ones

constructive and algebraic methods of the theory of rough sets this paper reviews and compares constructive and algebraic approaches in the study of rough sets in the constructive approach one starts from a binary relation and defines a pair of lower and upper approximation operators using the binary relation different classes of rough set algebras are obtained from different types of binary relations in the algebraic approach one defines a pair of dual approximation operators and states axioms that must be satisfied by the operators various classes of rough set algebras are characterized by different sets of axioms axioms of approximation operators guarantee the existence of certain types of binary relations producing the same operators

some conformally flat spin manifolds dirac operators and automorphic forms abstract in this paper we study clifford and harmonic analysis on some examples of conformal flat manifolds that have a spinor structure or more generally at least a pin structure the examples treated here are manifolds that can be parametrized by u where u is a subdomain of either s n or r n and is a kleinian group acting discontinuously on u the examples studied here include rp n and the hopf manifolds s 1 s n 1 also some hyperbolic manifolds will be treated special kinds of clifford analytic automorphic forms associated to the different choices of are used to construct explicit cauchy kernels cauchy integral formulas green s kernels and formulas together with hardy spaces and plemelj projection operators for l p spaces of hypersurfaces lying in these manifolds

a note on the contraction of lie algebras it is demonstrated that the contraction of lie algebras can be viewed as the procedure that underlies limiting distributions in probability theory the consequences of such an interpretation are discussed

relationship among basic concepts in covering based rough sets uncertainty and incompleteness of knowledge are widespread phenomena in information systems rough set theory is a tool for dealing with granularity and vagueness in data analysis rough set method has already been applied to various fields such as process control economics medical diagnosis biochemistry environmental science biology chemistry psychology and conflict analysis covering based rough set theory is an extension to classical rough sets in covering based rough sets there exist several basic concepts such as reducible elements of a covering minimal descriptions unary coverings and the property that the intersection of any two elements is the union of finite elements in this covering these concepts appeared in the literature of covering based rough sets separately in this paper we study the relationships between them in particular we establish the equivalence of the unary covering and the covering with the property that the intersection of any two elements is the union of finite elements in this covering we also investigate the relationship between the covering lower approximation operation and the interior operator a characterization of the interior operator by the covering lower approximation operation is presented in this paper correspondingly we study the relationship between the covering upper approximation operation and the closure operator in addition we explore the conditions under which the covering upper approximation operation is monotone the study of the relationships between these concepts will help us have a better understanding of covering based rough sets

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

covering based rough set approximations we propose a framework for the study of covering based rough set approximations three equivalent formulations of the classical rough sets are examined by using equivalence relations partitions and s algebras respectively they suggest the element based the granule based and the subsystem based definitions of approximation operators covering based rough sets are systematically investigated by generalizing these formulations and definitions a covering of universe of objects is used to generate different neighborhood operators neighborhood systems coverings and subsystems of the power set of the universe they are in turn used to define different types of generalized approximation operators within the proposed framework we review and discuss covering based approximation operators according to the element granule and subsystem based definitions

cost sensitive feature selection based on adaptive neighborhood granularity with multi level confidence neighborhood rough set model is considered as one of the effective granular computing models in dealing with numerical data this model is now widely discussed in feature selection and rule learning however there is no theoretical analysis on the issue of neighborhood granularity selection the influence of sampling resolution test and misclassification costs on modeling in this paper we design an adaptive neighborhood rough set model according to data precision and develop a fast backtracking algorithm for neighborhood rough sets based cost sensitive feature selection by considering the trade off between test costs and misclassification costs in the proposed model the neighborhood granularity based on the 3 rule of statistics is adaptive to data precision that is described by the multi level confidence of the feature subsets our experiments thoroughly performed on 12 datasets demonstrate the effectiveness of the model and the efficiency of the backtracking algorithm

rule learning for classification based on neighborhood covering reduction rough set theory has been extensively discussed in the domain of machine learning and data mining pawlak s rough set theory offers a formal theoretical framework for attribute reduction and rule learning from nominal data however this model is not applicable to numerical data which widely exist in real world applications in this work we extend this framework to numerical feature spaces by replacing partition of universe with neighborhood covering and derive a neighborhood covering reduction based approach to extracting rules from numerical data we first analyze the definition of covering reduction and point out its advantages and disadvantages then we introduce the definition of relative covering reduction and develop an algorithm to compute it given a feature space we compute the neighborhood of each sample and form a neighborhood covering of the universe and then employ the algorithm of relative covering reduction to the neighborhood covering thus derive a minimal covering rule set some numerical experiments are presented to show the effectiveness of the proposed technique

interpolating coherent states for heisenberg weyl and single photon su 1 1 algebras new quantal states which interpolate between the coherent states of the heisenberg weyl w3 and su 1 1 algebras are introduced the interpolating states are coherent states of a closed and symmetric algebra containing a parameter k which takes values from zero to one the elements of the algebra are realized in terms of operator valued functions of creation and annihilation operators the realization is such that the operators become those of the w3 algebra when k 0 and of the su 1 1 algebra when k is unity the overcompleteness of the interpolating coherent states is established by constructing suitable integration measures differential operator representations for the elements of the algebra are given along with the relevant spaces of entire functions a nonsymmetric set of operators to realize the w3 algebra is presented and the relevant coherent states are studied physical properties such as squeezing and photon statistics of the interpolating coherent states are investigated