representations of kac moody and virasoro algebras on modules of greenberg mohapatra algebras the theory of greenberg mohapatra algebras is developed in continuation to a previous paper standard cyclic modules are constructed for the algebras and on them kac moody and virasoro algebras are represented through a basic procedure the representations of kac moody and virasoro algebras help to understand the structure of greenberg mohapatra algebras and the question of possible violation of the pauli exclusion principle

lle score a new filter based unsupervised feature selection method based on nonlinear manifold embedding and its application to image recognition the task of feature selection is to find the most representative features from the original high dimensional data because of the absence of the information of class labels selecting the appropriate features in unsupervised learning scenarios is much harder than that in supervised scenarios in this paper we investigate the potential of locally linear embedding lle which is a popular manifold learning method in feature selection task it is straightforward to apply the idea of lle to the graph preserving feature selection framework however we find that this straightforward application suffers from some problems for example it fails when the elements in the feature are all equal it does not enjoy the property of scaling invariance and cannot capture the change of the graph efficiently to solve these problems we propose a new filter based feature selection method based on lle in this paper which is named as lle score the proposed criterion measures the difference between the local structure of each feature and that of the original data our experiments of classification task on two face image data sets an object image data set and a handwriting digits data set show that lle score outperforms state of the art methods including data variance laplacian score and sparsity score

the adler shiota van moerbeke formula for the bkp hierarchy we study the bkp hierarchy and prove the existence of an adler shiota van moerbeke formula this formula relates the action of the bw1 algebra on tau functions to the action of the additional symmetries on wave functions

branching rules for conformal embeddings we give explicit formulas for the branching rules of the conformal embeddingssu n n 1 2 1 su n n 2 su n n 1 2 1 su n n 2 sp n 1 so n 4 su 2 n andso m n 1 so m 1 so n 1 withm andn odd

learning a mahalanobis metric from equivalence constraints many learning algorithms use a metric defined over the input space as a principal tool and their performance critically depends on the quality of this metric we address the problem of learning metrics using side information in the form of equivalence constraints unlike labels we demonstrate that this type of side information can sometimes be automatically obtained without the need of human intervention we show how such side information can be used to modify the representation of the data leading to improved clustering and classification specifically we present the relevant component analysis rca algorithm which is a simple and efficient algorithm for learning a mahalanobis metric we show that rca is the solution of an interesting optimization problem founded on an information theoretic basis if dimensionality reduction is allowed within rca we show that it is optimally accomplished by a version of fisher s linear discriminant that uses constraints moreover under certain gaussian assumptions rca can be viewed as a maximum likelihood estimation of the within class covariance matrix we conclude with extensive empirical evaluations of rca showing its advantage over alternative methods

constrained large margin local projection algorithms and extensions for multimodal dimensionality reduction a constrained large margin local projection cmlp technique for multimodal dimensionality reduction is proposed we elaborate the criterion of cmlp from a pairwise constrained marginal perspective four effective cmlp solution schemes are presented and the corresponding comparative analyses are given an equivalent weighted least squares formulation for cmlp is also detailed cmlp is originated from the criterion of locality preserving projections lpp but cmlp offers a number of attractive advantages over lpp to keep the intrinsic proximity relations of inter class and intra class similarity pairs the localized pairwise cannot link and must link constraints are applied to specify the types of those neighboring pairs by utilizing the cmlp criterion margins between inter and intra class clusters are significantly enlarged as a result multimodal distributions are effectively preserved to further optimize the cmlp criterion one feasible improvement strategy is described with kernel methods we present the kernelized extensions of our approaches mathematical comparisons and analyses between this work and the related works are also detailed extensive simulations including multivariate manifold visualization and classification on the benchmark ucl orl yale umist mit cbcl and usps datasets are conducted to verify the efficiency of our techniques the presented results reveal that our methods are highly competitive with and even outperform some widely used state of the art algorithms

poisson lie group of pseudodifferential symbols we introduce a lie bialgebra structure on the central extension of the lie algebra of differential operators with scalar or matrix coefficients on the line and on the circle this defines a poisson lie structure on the dual group of pseudodifferential symbols of an arbitrary real or complex order we show that the usual second benney gl n kdv or gl n adler gelfand dickey and kp poisson structures are naturally realized as restrictions of this poisson structure to submanifolds of this universal poisson lie group moreover the reduced sl n versions of these manifolds orw n algebras in physical terminology can be viewed as certain subspaces of the quotient of this poisson lie group by the dressing action of the group of functions on the circle or as a result of a poisson reduction finally we define an infinite set of commuting functions on the poisson lie group that give the standard families of hamiltonians when restricted to the submanifolds mentioned above the poisson structure and hamiltonians on the whole group interpolate between the poisson structures and hamiltonians of benney kp and kdv flows we also discuss the geometrical meaning ofw as a limit of poisson algebraswe as e 0

feature evaluation and selection with cooperative game theory recent years various information theoretic based measurements have been proposed to remove redundant features from high dimensional data set as many as possible however most traditional information theoretic based selectors will ignore some features which have strong discriminatory power as a group but are weak as individuals to cope with this problem this paper introduces a cooperative game theory based framework to evaluate the power of each feature the power can be served as a metric of the importance of each feature according to the intricate and intrinsic interrelation among features then a general filter feature selection scheme is presented based on the introduced framework to handle the feature selection problem to verify the effectiveness of our method experimental comparisons with several other existing feature selection methods on fifteen uci data sets are carried out using four typical classifiers the results show that the proposed algorithm achieves better results than other methods in most cases

framed torsion free sheaves on cp2 hilbert schemes and representations of infinite dimensional lie algebras abstract we construct representations of both heiseberg and clifford algebras on the equivariant cohomology of the moduli space of framed torsion free sheaves on p 2 as well as on the equivariant cohomology of the hilbert scheme of points on the resolution of a type a simple singularity this generalizes the earlier work of nakajima grojnowski and others

spin and wedge representations of infinite dimensional lie algebras and groups abstract we suggest a purely algebraic construction of the spin representation of an infinite dimensional orthogonal lie algebra sections 1 and 2 and a corresponding group section 4 from this we deduce a construction of all level one highest weight representations of orthogonal affine lie algebras in terms of creation and annihilation operators on an infinite dimensional grassmann algebra section 3 we also give a similar construction of the level one representations of the general linear affine lie algebra in an infinite dimensional wedge space along these lines we construct the corresponding representations of the universal central extension of the group sln k t t 1 in spaces of sections of line bundles over infinite dimensional homogeneous spaces section 5

using powerdomains to generalize relational databases abstract much of relational algebra and the underlying principles of relational database design have a simple representation in the theory of domains that is traditionally used in the denotational semantics of programming languages by investigating the possible orderings on powerdomains that are well known in the study of nondeterminism and concurrency it is possible to show that many of the ideas in relational databases apply to structures that are much more general than relations this also suggests a method of representing database objects as typed objects in programming languages in this paper we show how operations such as natural join and projection which are fundamental to relational database design can be generalized and we use this generalized framework to give characterizations of several relational database concepts including functional dependencies and universal relations all of these have a simple minded semantics in terms of the underlying domains which can be thought of as domains of partial descriptions of real world objects we also discuss the applicability of relational database theory to nonrelational structures such as records with variants higher order relations recursive structures and other ordered spaces

the character of the infinite wedge representation abstract we study the character of the infinite wedge projective representation of the algebra of differential operators on the circle we prove quasi modularity of this character and also compute certain generating functions for traces of differential operators which we call correlation functions these correlation functions are sums of determinants built from genus 1 theta functions and their derivatives

maximum weight and minimum redundancy a novel framework for feature subset selection feature subset selection is often required as a preliminary work for many pattern recognition problems in this paper a novel filter framework is presented to select optimal feature subset based on a maximum weight and minimum redundancy mwmr criterion since the weight of each feature indicates its importance for some ad hoc tasks such as clustering and classification and the redundancy represents the correlations among features through the proposed mwmr we can select the feature subset in which the features are most beneficial to the subsequent tasks while the redundancy among them is minimal moreover a pair wise updating based iterative algorithm is introduced to solve our framework effectively in the experiments three feature weighting algorithms laplacian score fisher score and constraint score are combined with two redundancy measurement methods pearson correlation coefficient and mutual information to test the performances of proposed mwmr the experimental results on five different databases cmu pie extended yaleb colon dlbcl and pcmac demonstrate the advantage and efficiency of our mwmr

normalized cuts and image segmentation we propose a novel approach for solving the perceptual grouping problem in vision rather than focusing on local features and their consistencies in the image data our approach aims at extracting the global impression of an image we treat image segmentation as a graph partitioning problem and propose a novel global criterion the normalized cut for segmenting the graph the normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups we show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion we applied this approach to segmenting static images as well as motion sequences and found the results to be very encouraging

feature selection for unsupervised learning in this paper we identify two issues involved in developing an automated feature subset selection algorithm for unlabeled data the need for finding the number of clusters in conjunction with feature selection and the need for normalizing the bias of feature selection criteria with respect to dimension we explore the feature selection problem and these issues through fssem feature subset selection using expectation maximization em clustering and through two different performance criteria for evaluating candidate feature subsets scatter separability and maximum likelihood we present proofs on the dimensionality biases of these feature criteria and present a cross projection normalization scheme that can be applied to any criterion to ameliorate these biases our experiments show the need for feature selection the need for addressing these two issues and the effectiveness of our proposed solutions

a maximal monoidal closed category of distributive algebraic domains abstract we study the category bc of bounded complete dcpos and maps preserving all suprema linear maps bc is a symmetric monoidal closed category if sup denotes the full subcategory of bc with dcpos with one as objects we realize a categorical semantics of linear logic in sup the multiplicatives are fully distributive w r t the additives given prime the full subcategory of bc with prime algebraic dcpos as objects we introduce a prime algebraic quotient a which preserves all the logical operations in sup up to isomorphism therefore if c is a categorical semantics in bc then c is a categorical semantics in prime in particular sup the full subcategory of bc with all prime algebraic lattices as objects is such a categorical semantics prime is symmetric monoidal closed and maximal with respect to being closed under formula and if we demand that all objects are algebraic and distributive thus sup is a maximal categorical semantics with respect to these conditions we discuss the modalities and in sup

generalization of the algebra of the matrix pseudodifferential operators we introduce the new nonhomogeneous graded matrix algebra matn r over the ring r of the pseudodifferential operators of the arbitrary orders this algebra forms the generalization of the algebra of the standard matrix pseudodifferential operators we show that the matn r algebra admits the nontrivial central extension the case n 2 is considered in some detail as the example we consider the mat2 w1 mat2 virasoro and mat2 sl 2 c algebras

weight evaluation for features via constrained data pairscan t linkq facing the massive amount of data appearing on the web automatic analysis tools have become essential for web users to discover valuable information online precise similarity measurement plays a decisive role in enabling analysis tools to acquire high quality performances because different features contribute diversely to similarity calculation it is necessary to utilize weight to measure feature s contribution and import it into similarity measurement to accurately assign feature s weight constrained data pairs provided by users are usually imported into the weight evaluation procedure whereas conventional plans all fail to consider two challenges a asymmetrical distribution of constrained data pairs and b inconsistency contained by constrained data pairs if these two issues occur conventional plans are incompetent at addressing them or are even unable to work thus this paper proposes a novel constraint based weight evaluation to address these two issues for the former constrained data pairs are partitioned into several equivalent classes and distributing parameters are assigned to constrained data pairs to balance their distributions for the latter constrained data pairs are connected one after another and belief values are thereby formed to indicate their probability of being inconsistent experimental results demonstrate that this type of evaluation is independent of any algorithm with this evaluation similarities can be calculated more accurately

quasifinite representations of classical lie subalgebras of w p we show that there are exactly two anti involutions of the algebra of differential operators on the circle that are a multiple of p t t preserving the principal gradation p c x non constant we classify the irreducible quasifinite highest weight representations of the central extension dp of the lie subalgebra fixed by the most important cases are the subalgebras dx of w that are obtained when p x x in these cases we realize the irreducible quasifinite highest weight modules in terms of highest weight representation of the central extension of the lie algebra of infinite matrices with finitely many nonzero diagonals over the algebra c u um 1 and its classical lie subalgebras of c and d types

highest weight irreducible representations of the lie superalgebra gl 1 two classes of irreducible highest weight modules of the general linear lie superalgebra gl 1 are constructed within each module a basis is introduced and the transformation relations of the basis under the action of the algebra generators are written down

conformal embeddings rank level duality and exceptional modular invariants we compute the branching rules of the conformal embeddingsso 4nk 1 sp 2n k sp 2k n andso rq 1 so r q so q r forrq even using this we prove that the affine algebrassp 2n k andsp 2k n have the sames matrix and modular invariants as a second application we show how the triality ofso 8 leads to an exceptional modular invariant forsu 2 at level 16 and for allso q 4 at level 8

harish chandra modules over the twisted heisenberg virasoro algebra in this paper we classify all indecomposable harish chandra modules of the intermediate series over the twisted heisenberg virasoro algebra meanwhile some bosonic modules are also studied

a decomposition theorem for domains a domain constructor that generalizes the product is defined it is shown that with this constructor exactly the prime algebraic coherent scott domains can be generated from two chains and boolean flat domains

constraint score a new filter method for feature selection with pairwise constraints feature selection is an important preprocessing step in mining high dimensional data generally supervised feature selection methods with supervision information are superior to unsupervised ones without supervision information in the literature nearly all existing supervised feature selection methods use class labels as supervision information in this paper we propose to use another form of supervision information for feature selection i e pairwise constraints which specifies whether a pair of data samples belong to the same class must link constraints or different classes cannot link constraints pairwise constraints arise naturally in many tasks and are more practical and inexpensive than class labels this topic has not yet been addressed in feature selection research we call our pairwise constraints guided feature selection algorithm as constraint score and compare it with the well known fisher score and laplacian score algorithms experiments are carried out on several high dimensional uci and face data sets experimental results show that with very few pairwise constraints constraint score achieves similar or even higher performance than fisher score with full class labels on the whole training data and significantly outperforms laplacian score

the logarithm of the derivative operator and higher spin algebras of w type we use the notion of the logarithm of the derivative operator to describew type algebras as central extensions of the algebra of differential operators we also provide closed formulae for the truncations ofw1 to higher spin algebras withs m for allm 2 the results are extended to matrix valued differential operators introducing a logarithmic generalization of the maurer cartan cocycle

dual representations of and decomposition of fock spaces in this paper we study left and right regular representations of an infinite dimensional analogue of the classical lie group with entries indexed by the integers on a space which we take to be the projective limit of bargmann fock segal spaces we decompose into an orthogonal direct sum where the hat indicates the closure of the algebraic direct sum and where each is irreducible with respect to the joint left and right action and the isotypic component of the left or right actions with double signatures of the type for each is the subspace generated by the joint action on a highest weight vector with highest weight we start by defining the group and its lie algebra and discuss the infinite wedge space f defined by kac and its transpose we then define a kind of determinant function on prove several identities and properties of these determinants and use these to produce intertwining maps from tensor products of f and to thereby obtaining irreducible subspaces and highest weight vectors we further adapt these results to representations of on the inductive limit of bargmann fock segal spaces

semi infinite weil complex and the virasoro algebra we define a semi infinite analogue of the weil algebra associated an infinite dimensional lie algebra it can be used for the definition of semi infinite characteristic classes by analogy with the chern weil construction the second term of a spectral sequence of this weil complex consists of the semi infinite cohomology of the lie algebra with coefficients in its adjoint semi infinite symmetric powers we compute this cohomology for the virasoro algebra this is just the brst cohomology of the bosonic system with central charge 26 we give a complete description of the fock representations of this bosonic system as modules over the virasoro algebra using friedan martinec shenker bosonization we derive a combinatorial identity from this result

decomposition rules for conformal pairs associated to symmetric spaces and abelian subalgebras of z2 graded lie algebras we give uniform formulas for the branching rules of level 1 modules over orthogonal affine lie algebras for all conformal pairs associated to symmetric spaces we also provide a combinatorial interpretation of these formulas in terms of certain abelian subalgebras of simple lie algebras

unsupervised feature selection by regularized self representation by removing the irrelevant and redundant features feature selection aims to find a compact representation of the original feature with good generalization ability with the prevalence of unlabeled data unsupervised feature selection has shown to be effective in alleviating the curse of dimensionality and is essential for comprehensive analysis and understanding of myriads of unlabeled high dimensional data motivated by the success of low rank representation in subspace clustering we propose a regularized self representation rsr model for unsupervised feature selection where each feature can be represented as the linear combination of its relevant features by using l 2 1 norm to characterize the representation coefficient matrix and the representation residual matrix rsr is effective to select representative features and ensure the robustness to outliers if a feature is important then it will participate in the representation of most of other features leading to a significant row of representation coefficients and vice versa experimental analysis on synthetic and real world data demonstrates that the proposed method can effectively identify the representative features outperforming many state of the art unsupervised feature selection methods in terms of clustering accuracy redundancy reduction and classification accuracy highlightsa regularized self representation rsr model is proposed for unsupervised feature selection an iterative reweighted least squares algorithm is proposed to solve the rsr model the proposed method shows superior performance to state of the art

tau functions as highest weight vectors for algebra for each we construct a highest weight module of the lie algebra the highest weight vectors are specific tau functions of the nth gelfand dickey hierarchy we show that these modules are quasifinite and we give a complete description of the reducible ones together with a formula for the singular vectors

petri nets event structures and domains part i abstract the general aim of this paper is to find a theory of concurrency combining the approaches of petri and scott and others in part i we introduce our formalisms to connect the abstract ideas of events and domains of information we show how casual nets induce certain kinds of domains where the information points are certain sets of events this allows translations between the languages of net theory and domain theory following the idea that events of causal nets are occurrences we generalise causal nets to occurrence nets by adding forwards conflict just as infinite flow charts unfold finite ones so transition nets can be unfolded into occurrence nets next we extend the above connections between nets and domains to these new nets event structures which are intermediate between nets and domains play an important part in all our work finally as an example of how concepts translate from one formalism to the other we show how petri s notion of confusion ties up with kahn and plotkin s concrete domains in part ii we shall continue the job of connecting up notions within net theory and the theory of domains in particular we shall examine the idea of states of computations

bagging constraint score for feature selection with pairwise constraints constraint score is a recently proposed method for feature selection by using pairwise constraints which specify whether a pair of instances belongs to the same class or not it has been shown that the constraint score with only a small amount of pairwise constraints achieves comparable performance to those fully supervised feature selection methods such as fisher score however one major disadvantage of the constraint score is that its performance is dependent on a good selection on the composition and cardinality of constraint set which is very challenging in practice in this work we address the problem by importing bagging into constraint score and a new method called bagging constraint score bcs is proposed instead of seeking one appropriate constraint set for single constraint score in bcs we perform multiple constraint score each of which uses a bootstrapped subset of original given constraint set diversity analysis on individuals of ensemble shows that resampling pairwise constraints is helpful for simultaneously improving accuracy and diversity of individuals we conduct extensive experiments on a series of high dimensional datasets from uci repository and gene databases and the experimental results validate the effectiveness of the proposed method

novel soft subspace clustering with multi objective evolutionary approach for high dimensional data many conventional soft subspace clustering techniques merge several criteria into a single objective to improve performance however the weighting parameters become important but difficult to set in this paper a novel soft subspace clustering with a multi objective evolutionary approach moeassc is proposed to this problem this clustering method considers two types of criteria as multiple objectives and optimizes them simultaneously by using a modified multi objective evolutionary algorithm with new encoding and operators an indicator called projection similarity validity index psvindex is designed to select the best solution and cluster number experiments on many datasets demonstrate the usefulness of moeassc and psvindex and show that our algorithm is insensitive to its parameters and is scalable to large datasets

matrixized learning machine with modified pairwise constraints matrix pattern oriented classifier design matcd has been demonstrated to be effective in terms of the classification performance since it utilizes two sided weight vectors to constrain the matrix based patterns however the existing matcd might not be able to acquire the prior distribution knowledge such as the relationship between two patterns inspired by the pairwise constraints pc method i e must links and cannot links between the patterns this paper introduces a new regularization term named rp with a modified pc method into matcd the new classifier design strategy is expected to not only learn the structural information of each pattern itself but also acquire the prior distribution knowledge about each constrained pair with both the discrimination metric from the traditional pc and the spatial distance measure from the heat kernel method in practice this paper selects one typical matrixized classifier named matmhks as the basic building block and introduces the term rp into it the newly proposed classifier is named mlmmpc and the subsequent experiments validate the effectiveness of it two major contributions of this paper can be concluded as 1 improving the existing matrix pattern oriented classifier design techniques and 2 modifying the traditional pc method by combining the discrimination metric and the distance measure together highlightsa novel matrix oriented classification algorithm named mlmmpc is proposed to improve the original matrix learning framework by a new regularization term r p to combine pairwise constraints and spatial measure together in r p implementability is demonstrated on both image and vector based datasets

bcn graded lie algebras arising from fermionic representations abstract we use fermionic representations to obtain a class of bc n graded lie algebras coordinatized by quantum tori with nontrivial central extensions

extensions and contractions of the lie algebra ofq pseudodifferential symbols on the circle abstract we construct cocycles on the lie algebra of pseudo and q pseudodifferential symbols of one variable and on their close relatives the sine algebra and the poisson algebra on two torus a quantum godbillon vey cocycle on pseudo differential operators appears in this construction as a natural generalization of the gelfand fuchs 3 cocycle on periodic vector fields a nontrivial embedding of the virasoro algebra into a completion of q pseudodifferential symbols is proposed we describe q analogs of the kp and kdv hierarchies admitting an infinite number of conserved charges as well as q deformed gelfand dickey structures

svm fuzcoc a novel svm based feature selection method using a fuzzy complementary criterion an efficient filter feature selection fs method is proposed in this paper the svm fuzcoc approach achieving a satisfactory trade off between classification accuracy and dimensionality reduction additionally the method has reasonably low computational requirements even in high dimensional feature spaces to assess the quality of features we introduce a local fuzzy evaluation measure with respect to patterns that embraces fuzzy membership degrees of every pattern in their classes accordingly the above measure reveals the adequacy of data coverage provided by each feature the required membership grades are determined via a novel fuzzy output kernel based support vector machine applied on single features based on a fuzzy complementary criterion fuzcoc the fs procedure iteratively selects features with maximum additional contribution in regard to the information content provided by previously selected features this search strategy leads to small subsets of powerful and complementary features alleviating the feature redundancy problem we also devise different svm fuzcoc variants by employing seven other methods to derive fuzzy degrees from svm outputs based on probabilistic or fuzzy criteria our method is compared with a set of existing fs methods in terms of performance capability dimensionality reduction and computational speed via a comprehensive experimental setup including synthetic and real world datasets

quantum field theories in one and two dimensions axiomes pour les theories de champ quantiques theories quantiques du champ a une dimension theories quantiques du champ chirales a 2 dimensions

improving constrained clustering with active query selection in this article we address the problem of automatic constraint selection to improve the performance of constraint based clustering algorithms to this aim we propose a novel active learning algorithm that relies on a k nearest neighbors graph and a new constraint utility function to generate queries to the human expert this mechanism is paired with propagation and refinement processes that limit the number of constraint candidates and introduce a minimal diversity in the proposed constraints existing constraint selection heuristics are based on a random selection or on a min max criterion and thus are either inefficient or more adapted to spherical clusters contrary to these approaches our method is designed to be beneficial for all constraint based clustering algorithms comparative experiments conducted on real datasets and with two distinct representative constraint based clustering algorithms show that our approach significantly improves clustering quality while minimizing the number of human expert solicitations

exploring the boundary region of tolerance rough sets for feature selection of all of the challenges which face the effective application of computational intelligence technologies for pattern recognition dataset dimensionality is undoubtedly one of the primary impediments in order for pattern classifiers to be efficient a dimensionality reduction stage is usually performed prior to classification much use has been made of rough set theory for this purpose as it is completely data driven and no other information is required most other methods require some additional knowledge however traditional rough set based methods in the literature are restricted to the requirement that all data must be discrete it is therefore not possible to consider real valued or noisy data this is usually addressed by employing a discretisation method which can result in information loss this paper proposes a new approach based on the tolerance rough set model which has the ability to deal with real valued data whilst simultaneously retaining dataset semantics more significantly this paper describes the underlying mechanism for this new approach to utilise the information contained within the boundary region or region of uncertainty the use of this information can result in the discovery of more compact feature subsets and improved classification accuracy these results are supported by an experimental evaluation which compares the proposed approach with a number of existing feature selection techniques

generalized theta functions strange duality and odd orthogonal bundles on curves this paper studies spaces of generalized theta functions for odd orthogonal bundles with nontrivial stiefel whitney class and the associated space of twisted spin bundles in particular we prove a verlinde type formula and a dimension equality that was conjectured by oxbury wilson modifying hitchin s argument we also show that the bundle of generalized theta functions for twisted spin bundles over the moduli space of curves admits a flat projective connection we furthermore address the issue of strange duality for odd orthogonal bundles and we demonstrate that the naive conjecture fails in general a consequence of this is the reducibility of the projective representations of spin mapping class groups arising from the hitchin connection for these moduli spaces finally we answer a question of nakanishi tsuchiya about rank level duality for conformal blocks on the pointed projective line with spin weights

cost sensitive rough set approach we consider test cost and decision cost simultaneously cost sensitive rough set was proposed and explored attribute reductions based on three different criteria were investigated cost sensitivity is an important problem which has been addressed by many researchers around the world as far as cost sensitivity in the rough set theory is concerned two types of important costs have been seriously considered on the one hand the decision cost has been introduced into the modeling of decision theoretic rough set on the other hand the test cost has been taken into account in attribute reduction however few researchers pay attention to the construction of rough set model with test cost and decision cost simultaneously to fill such a gap a new cost sensitive rough set approach is proposed in which the information granules are sensitive to test costs and approximations are sensitive to decision costs respectively furthermore with respect to different criteria of positive region preservation decision monotonicity and cost decrease three heuristic algorithms are designed to compute reducts respectively the comparisons among these three algorithms show us 1 positive region preservation based algorithm can keep the decision rules supported by lower approximation region unchanged 2 decision monotonicity based heuristic algorithm can obtain a reduct with more positive decision rules and higher classification accuracy 3 cost minimum based algorithm can generate a reduct with minor cost this study suggests potential application areas and new research trends concerning rough set theory