mckay correspondence for symplectic quotient singularities let v be a finite dimensional complex vector space and g sl v a finite subgroup the quotient variety x v g is usually singular if dim v 2 then there exists a canonical smooth projective resolution y x of the singular variety x this resolution is crepant which in this case means that the manifold y has trivial canonical bundle we recall the precise definitions in definition 1 2 the fiber 1 0 y over the singular point 0 x v g is a rational curve whose components are numbered by the nontrivial conjugacy classes in g the homology classes of these components freely generate h2 y q all the other homology groups of y are trivial except for h0 y q q this situation was described by j mckay in mck it is known as the mckay correspondence in higher dimensions the picture splits into two parts one can consider two separate questions 1 when does the quotient x v g admit a smooth crepant resolution y x 2 assuming there exists a smooth crepant resolution y x what can one say about the homology h y q

on the homology of the hilbert scheme of points in the plane geir ellingsrud 1 and stein arild stromme 2 i matematisk institutt universitetet i oslo blindern n oslo 3 norway 2 matematisk institutt universitetet i bergen n 5014 bergen norway although several authors have been interested in the hilbert scheme hilba ip z parametrizing finite subschemes of length d in the projective plane i1 i2 f 1 f2 br among others not much is known about the topological properties of this space the picard group has been calculated i f2 and the homology groups of hilb3 ip 2 have been computed hi in this paper we give a precise description of the additive structure of the homology of hilba ip2 applying the results of birula bialynicki b1 b2 on the cellular decompositions de fined by a torus action to the natural action of a maximal torus of sl 3 on hilbe ipz a rather easy consequence of the fact that this action has finitely many fixpoints is that the cycle maps between the chow groups and the homology groups are isomorphisms in particular there is no odd homology and the homology groups are all free the main objective of this work is to compute their ranks the betti numbers of hilba ipz as a byproduct of our method we get similar results on the homology of the punctual hilbert scheme and of the hilbert scheme of points in the affine plane it seems natural to generalize our results to any toric smooth surface however we give the results only for the rational ruled surfaces if with an indication of the necessary changes in the proofs for simplicity we work over the field of complex numbers but with an appropriate interpretation of the word homology our results remain valid over any base field

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

a study on relationship between generalization abilities and fuzziness of base classifiers in ensemble learning we investigate essential relationships between generalization capabilities and fuzziness of fuzzy classifiers viz the classifiers whose outputs are vectors of membership grades of a pattern to the individual classes the study makes a claim and offers sound evidence behind the observation that higher fuzziness of a fuzzy classifier may imply better generalization aspects of the classifier especially for classification data exhibiting complex boundaries this observation is not intuitive with a commonly accepted position in traditional pattern recognition the relationship that obeys the conditional maximum entropy principle is experimentally confirmed furthermore the relationship can be explained by the fact that samples located close to classification boundaries are more difficult to be correctly classified than the samples positioned far from the boundaries this relationship is expected to provide some guidelines as to the improvement of generalization aspects of fuzzy classifiers

reflection of transversal progressing waves in nonlinear strictly hyperbolic mixed problems etant donnees des solutions suffisamment lisses d un probleme hyperbolique mixte semilineaire on montre qu une onde progressive qui frappe la frontiere est reflechie selon la loi usuelle

progressive subspace ensemble learning there are not many classifier ensemble approaches which investigate the data sample space and the feature space at the same time and this multi pronged approach will be helpful for constructing more powerful learning models for example the adaboost approach only investigates the data sample space while the random subspace technique only focuses on the feature space to address this limitation we propose the progressive subspace ensemble learning approach psel which takes into account the data sample space and the feature space at the same time specifically psel first adopts the random subspace technique to generate a set of subspaces then a progressive selection process based on new cost functions that incorporate current and long term information to select the classifiers sequentially will be introduced finally a weighted voting scheme is used to summarize the predicted labels and obtain the final result we also adopt a number of non parametric tests to compare psel and its competitors over multiple datasets the results of the experiments show that psel works well on most of the real datasets and outperforms a number of state of the art classifier ensemble approaches highlightsprogressive subspace ensemble learning it takes into account the data sample space and the feature space at the same time a progressive selection process based on new cost functions that incorporate current and long term information to select the classifiers sequentially will be introduced

a new generalisation of macdonald polynomials we introduce a new family of symmetric multivariate polynomials whose coefficients are meromorphic functions of two parameters q t and polynomial in a further two parameters u v we evaluate these polynomials explicitly as a matrix product at u v 0 they reduce to macdonald polynomials while at q 0 u v s they recover a family of inhomogeneous symmetric functions originally introduced by borodin

attribute bagging improving accuracy of classifier ensembles by using random feature subsets we present attribute bagging ab a technique for improving the accuracy and stability of classifier ensembles induced using random subsets of features ab is a wrapper method that can be used with any learning algorithm it establishes an appropriate attribute subset size and then randomly selects subsets of features creating projections of the training set on which the ensemble classifiers are built the induced classifiers are then used for voting this article compares the performance of our ab method with bagging and other algorithms on a hand pose recognition dataset it is shown that ab gives consistently better results than bagging both in accuracy and stability the performance of ensemble voting in bagging and the ab method as a function of the attribute subset size and the number of voters for both weighted and unweighted voting is tested and discussed we also demonstrate that ranking the attribute subsets by their classification accuracy and voting using only the best subsets further improves the resulting performance of the ensemble

the mckay correspondence as an equivalence of derived categories the classical mckay correspondence relates representations of a finite subgroup g sl 2 c to the cohomology of the well known minimal resolution of the kleinian singularity c2 g gonzalez sprinberg and verdier 10 interpreted the mckay correspondence as an isomorphism on k theory observing that the representation ring of g is equal to the g equivariant k theory of c2 more precisely they identify a basis of the k theory of the resolution consisting of the classes of certain tautological sheaves associated to the irreducible representations of g

ideals of quasi symmetric functions and super covariant polynomials for sn the aim of this work is to study the quotient ring r n of the ring q x 1 x n over the ideal j n generated by non constant homogeneous quasi symmetric functions we prove here that the dimension of r n is given by c n the n th catalan number this is also the dimension of the space sh n of super covariant polynomials that is defined as the orthogonal complement of j n with respect to a given scalar product we construct a basis for r n whose elements are naturally indexed by dyck paths this allows us to understand the hilbert series of sh n in terms of number of dyck paths with a given number of factors

selected tree classifier combination based on both accuracy and error diversity this paper proposes a method for combining multiple tree classifiers based on both classifier ensemble bagging and dynamic classifier selection schemes dcs the proposed method is composed of the following procedures 1 building individual tree classifiers based on bootstrap samples 2 calculating the distance between all possible two trees 3 clustering the trees based on single linkage clustering 4 selecting two clusters by local region in terms of accuracy and error diversity and 5 voting the results of tree classifiers selected in the two clusters empirical evaluation using publicly available data sets confirms the superiority of our proposed approach over other classifier combining methods

remarks on the derived mckay correspondence for hilbert schemes of points and tautological bundles we study the images of tautological bundles on hilbert schemes of points on surfaces and their wedge powers under the derived mckay correspondence the main observation of the paper is that using a derived equivalence differing slightly from the standard one considerably simplifies both the results and their proofs as an application we obtain shorter proofs for known results as well as new formulae for homological invariants of tautological sheaves in particular we compute the extension groups between wedge powers of tautological bundles associated to line bundles on the surface

the model of fuzzy variable precision rough sets the fuzzy rough set frs model has been introduced to handle databases with real values however frs was sensitive to misclassification and perturbation here misclassification means error or missing values in classification and perturbation means small changes of numerical data the variable precision rough sets vprss model was introduced to handle databases with misclassification however it could not effectively handle the real valued datasets now it is valuable from theoretical and practical aspects to combine frs and vprs so that a powerful tool which not only can handle numerical data but also is less sensitive to misclassification and perturbation can be developed in this paper we set up a model named fuzzy vprss fvprss by combining frs and vprs with the goal of making frs a special case first we study the knowledge representation ways of frs and vprs and then propose the set approximation operators of fvprs second we employ the discernibility matrix approach to investigate the structure of attribute reductions in fvprs and develop an algorithm to find all reductions third in order to overcome the np complete problem of finding all reductions we develop some fast heuristic algorithms to obtain one near optimal attribute reduction finally we compare fvprs with rs frs and several flexible rs based approaches with respect to misclassification and perturbation the experimental comparisons show the feasibility and effectiveness of fvprs

a novel dynamic rough subspace based selective ensemble ensemble learning has been a hot topic in machine learning due to its successful utilization in many applications rough set theory has been proved to be an excellent mathematical tool for dimension reduction in this paper based on rough set a novel framework for ensemble is proposed in our proposed framework the relationship among attributes in rough subspace is first considered and the maximum dependency degree of attribute is first employed to effectively reduce the searching space of reducts and augment the diversity of selected reducts in addition in order to choose an appropriate reduct from the dynamic reduct searching space an assessment function which can balance the accuracy and diversity is utilized at last a new method i e dynamic rough subspace based selective ensemble drsse which is derived from our framework is given by repeatedly changing the searching space of reducts and selecting the next reduct from the changed searching space drsse finally trains an ensemble system with these selected reducts compared with several available ensemble methods experimental results with several datasets demonstrate that drsse can lead to a comparative or even better performance a new framework for rough set ensemble and algorithm drsse is proposed dynamic searching space is used to increase the diversity of rough subspaces the relationship among attributes is considered to reduce the searching space consider the accuracy and diversity of base classifiers in an ensemble system

a graded representation model for macdonald s polynomials we define doubly graded sn modules rmu for which we conjecture that the multiplicities of irreducible representations in various bi degrees are given by the macdonald coefficients klambdamu assuming one fundamental conjecture the modules rmu can be given several equivalent definitions which we discuss we prove the conjectures in various special cases

vertex operators grassmannians and hilbert schemes we approximate the infinite grassmannian by finite dimensional cutoffs and define a family of fermionic vertex operators as the limit of geometric correspondences on the equivariant cohomology groups with respect to a one dimensional torus action we prove that in the localization basis these are the well known fermionic vertex operators on the infinite wedge representation furthermore the boson fermion correspondence locality and intertwining properties with the virasoro algebra are the limits of relations on the finite dimensional cutoff spaces which are true for geometric reasons we then show that these operators are also almost by definition the vertex operators defined by okounkov and the author in carlsson and okounkov http arxiv org abs 0801 2565v2 math ag 2009 on the equivariant cohomology groups of the hilbert scheme of points on mathbb c 2 with respect to a special torus action

the singularities of the principal component of the hilbert scheme of points abstract haiman proved the remarkable result that the isospectral hilbert scheme of points in the plane is normal cohen macaulay and gorenstein this implies the n conjecture and the positivity conjecture for the kostka macdonald coefficients in addition he conjectured that the isospectral hilbert scheme over the principal component of hilb n c d is cohen macaulay for any d n 1 we provide a counterexample to the conjecture

eros ensemble rough subspaces ensemble learning is attracting much attention from pattern recognition and machine learning domains for good generalization both theoretical and experimental researches show that combining a set of accurate and diverse classifiers will lead to a powerful classification system an algorithm called fs pp eros for selective ensemble of rough subspaces is proposed in this paper rough set based attribute reduction is introduced to generate a set of reducts and then each reduct is used to train a base classifier we introduce an accuracy guided forward search and post pruning strategy to select part of the base classifiers for constructing an efficient and effective ensemble system the experiments show that classification accuracies of ensemble systems with accuracy guided forward search strategy will increase at first arrive at a maximal value then decrease in sequentially adding the base classifiers we delete the base classifiers added after the maximal accuracy the experimental results show that the proposed ensemble systems outperform bagging and random subspace methods in terms of accuracy and size of ensemble systems fs pp eros can keep or improve the classification accuracy with very few base classifiers which leads to a powerful and compact classification system

on certain graded sn modules and the q kostka polynomials abstract we derive here a number of properties of the q kostka polynomials k q in particular we obtain a very accessible proof that these polynomials have non negative integer coefficients other monotonicity properties are also derived these results are obtained by studying certain graded s n modules r which afford a character that may be expressed in terms of the k q certain nesting properties of the r which correspond to the dominance order of partitions then translate themselves into combinatorial inequalities involving the k q the modules r have been given an elementary presentation by deconcini and procesi invent math 64 1981 203 219 as rather simple quotients of the polynomial ring q x 1 x 2 x n we show here that their basic properties may also be derived in an entirely elementary manner

double affine hecke algebras and macdonald s conjectures in this paper we prove the main results about the structure of double affine hecke algebras announced in c1 c2 the technique is based on the realization of these algebras in terms of demazure lusztig operators bgg d l2 ls c3 and rather standard facts from the theory of affine weyl groups in particular it completes the proof partially published in c2 of the macdonald scalar product conjecture see ml 12 6 including the famous macdonald constant term conjecture the q t case we mainly follow the opdam paper 0 where the macdonald mehta conjectures in the degenerate differential case were deduced from certain properties of the heckman opdam operators ho and the existence of the shift operators heckman s interpretation of these operators via the so called dunkl operators see he and also c5 was important to our approach we note that the ho operators are closely related to the so called quantum many body problem calogero sutherland moser olshanetsky perelomov the conformal field theory knizhnik zamolodchikov equations the harmonic analysis on symmetric spaces harish chandra helgason etc and last but not the least the classic theory of the hypergeometric functions establishing the connection between the difference counterparts of heckman opdam operators introduced in c4 and the macdonald theory ml m2 including the construction of the difference shift operators is the main thrust of this paper once the connection is established it is not very difficult to calculate the scalar squares of the macdonald polynomials and to prove the constant term conjecture from his fundamental paper m3 to simplify the exposition we discuss the reduced root systems only and impose the relation q tk for k e z to avoid infinite products in the definition of macdonald s pairing the purpose of this work is to present a concrete application of the new technique arbitrary q t can be handled in a

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

the random subspace method for constructing decision forests much of previous attention on decision trees focuses on the splitting criteria and optimization of tree sizes the dilemma between overfitting and achieving maximum accuracy is seldom resolved a method to construct a decision tree based classifier is proposed that maintains highest accuracy on training data and improves on generalization accuracy as it grows in complexity the classifier consists of multiple trees constructed systematically by pseudorandomly selecting subsets of components of the feature vector that is trees constructed in randomly chosen subspaces the subspace method is compared to single tree classifiers and other forest construction methods by experiments on publicly available datasets where the method s superiority is demonstrated we also discuss independence between trees in a forest and relate that to the combined classification accuracy