clustering high dimensional data a graph based relaxed optimization approach there is no doubt that clustering is one of the most studied data mining tasks nevertheless it remains a challenging problem to solve despite the many proposed clustering approaches graph based approaches solve the clustering task as a global optimization problem while many other works are based on local methods in this paper we propose a novel graph based algorithm gbr that relaxes some well defined method even as improving the accuracy whilst keeping it simple the primary motivation of our relaxation of the objective is to allow the reformulated objective to find well distributed cluster indicators for complicated data instances this relaxation results in an analytical solution that avoids the approximated iterative methods that have been adopted in many other graph based approaches the experiments on synthetic and real data sets show that our relaxation accomplishes excellent clustering results our key contributions are 1 we provide an analytical solution to solve the global clustering task as opposed to approximated iterative approaches 2 a very simple implementation using existing optimization packages 3 an algorithm with relatively less computation time over the number of data instances to cluster than other well defined methods in the literature

quantized knizhnik zamolodchikov equations quantum yang baxter equation and difference equations for q hypergeometric functions thesl2 quantized knizhnik zamolodchikov equations are solved inq hypergeometric functions new difference equations are derived for generalq hypergeometric functions the equations are given in terms of quantum yang baxter matrices and have the form similar to quantum knizhnik zamolodchikov equations for quantum affine algebras introduced by frenkel and reshetikhin

types of arcs in a fuzzy graph the concept of connectivity plays an important role in both theory and applications of fuzzy graphs depending on the strength of an arc this paper classifies arcs of a fuzzy graph into three types namely a strong b strong and d arcs the advantage of this type of classification is that it helps in understanding the basic structure of a fuzzy graph completely we analyze the relation between strong paths and strongest paths in a fuzzy graph and obtain characterizations for fuzzy bridges fuzzy trees and fuzzy cycles using the concept of a strong b strong and d arcs an arc of a fuzzy tree is a strong if and only if it is an arc of its unique maximum spanning tree also we identify different types of arcs in complete fuzzy graphs

fock space representations of affine lie algebras and integral representations in the wess zumino witten models fock space representations of affine lie algebras are studied explicit forms of correction terms adding to the currentsf i z are determined it is proved that the sugawara energy momentum tensor on the fock spaces is quadratic in free bosons furthermore screening operators are constructed this implies the existence of generalized hypergeometric integrals satisfying the knizhnik zamolodchikov equation

one class classifiers based on entropic spanning graphs one class classifiers offer valuable tools to assess the presence of outliers in data in this paper we propose a design methodology for one class classifiers based on entropic spanning graphs our approach also takes into account the possibility to process nonnumeric data by means of an embedding procedure the spanning graph is learned on the embedded input data and the outcoming partition of vertices defines the classifier the final partition is derived by exploiting a criterion based on mutual information minimization here we compute the mutual information by using a convenient formulation provided in terms of the alpha jensen difference once training is completed in order to associate a confidence level with the classifier decision a graph based fuzzy model is constructed the fuzzification process is based only on topological information of the vertices of the entropic spanning graph as such the proposed one class classifier is suitable also for data characterized by complex geometric structures we provide experiments on well known benchmarks containing both feature vectors and labeled graphs in addition we apply the method to the protein solubility recognition problem by considering several representations for the input samples experimental results demonstrate the effectiveness and versatility of the proposed method with respect to other state of the art approaches

manifold regularization a geometric framework for learning from labeled and unlabeled examples we propose a family of learning algorithms based on a new form of regularization that allows us to exploit the geometry of the marginal distribution we focus on a semi supervised framework that incorporates labeled and unlabeled data in a general purpose learner some transductive graph learning algorithms and standard methods including support vector machines and regularized least squares can be obtained as special cases we use properties of reproducing kernel hilbert spaces to prove new representer theorems that provide theoretical basis for the algorithms as a result in contrast to purely graph based approaches we obtain a natural out of sample extension to novel examples and so are able to handle both transductive and truly semi supervised settings we present experimental evidence suggesting that our semi supervised algorithms are able to use unlabeled data effectively finally we have a brief discussion of unsupervised and fully supervised learning within our general framework

on quantum separation of variables we present a new approach to construct the separate variables basis leading to the full characterization of the transfer matrix spectrum of quantum integrable lattice models the basis is generated by the repeated action of the transfer matrix itself on a generically chosen state of the hilbert space the fusion relations for the transfer matrix stemming from the yang baxter algebra properties provide the necessary closure relations to define the action of the transfer matrix on such a basis in terms of elementary local shifts leading to a separate transfer matrix spectral problem hence our scheme extends to the quantum case a key feature of the liouville arnold classical integrability framework where the complete set of conserved charges defines both the level manifold and the flows on it leading to the construction of action angle variables we work in the framework of the quantum inverse scattering method as a first example of our approach we give the construction of such a basis for models associated with y gln and argue how it extends to their trigonometric and elliptic versions then we show how our general scheme applies concretely to fundamental models associated with the y gl2 and y gl3 r matrices leading to the full characterization of their spectrum for y gl2 and its trigonometric deformation a particular case of our method reproduces sklyanin s construction of separate variables for y gl3 it gives new results in particular through the proper identification of the shifts acting on the separate basis we stress that our method also leads to the full characterization of the spectrum of other known quantum integrable lattice models including in particular trigonometric and elliptic spin chains open chains with general integrable boundaries and further higher rank cases that we will describe in forthcoming publications

integral representation for the eigenstates of the spin system with inverse square interactions we consider the calogero sutherland moser system with spin 1 2 we give the integral representation for the ground state wavefunction excited states of the calogero spin model confined in the harmonic potential are also given

linear neighborhood propagation and its applications in this paper a novel graph based transductive classification approach called linear neighborhood propagation is proposed the basic idea is to predict the label of a data point according to its neighbors in a linear way this method can be cast into a second order intrinsic gaussian markov random field framework its result corresponds to a solution to an approximate inhomogeneous biharmonic equation with dirichlet boundary conditions different from existing approaches our approach provides a novel graph structure construction method by introducing multiple wise edges instead of pairwise edges and presents an effective scheme to estimate the weights for such multiple wise edges to the best of our knowledge these two contributions are novel for semi supervised classification the experimental results on image segmentation and transductive classification demonstrate the effectiveness and efficiency of the proposed approach

spaces of quasi exponentials and representations of gln we consider the action of the bethe algebra on the weight subspace of weight of the tensor product of k polynomial irreducible modules with highest weights respectively the bethe algebra depends on n complex numbers under the assumption that k1 kn are distinct we prove that the image of in end is isomorphic to the algebra of functions on the intersection of suitable schubert cycles in the grassmannian of n dimensional spaces of quasi exponentials with exponents we also prove that the module is isomorphic to the coregular representation of that algebra of functions we present a bethe ansatz construction identifying the eigenvectors of the bethe algebra with points of that intersection of schubert cycles

a regularization framework in polar coordinates for transductive learning in networked data in networked data linked objects tend to belong to the same class and densely linked subgraphs are often available based on these facts this paper presents a regularization framework that consists of fitting and regularization terms for transductive learning in networked data the desirable value of the fitting term is related to the number of labeled data whereas that of the regularization term is dependent on the structure of the graph the ratio of these two desirable values is essential for the estimation of the optimal regularization parameters such as that proposed in our paper under the proposed regularization framework an effective classification algorithm is developed two methods are also introduced to incorporate contents of objects into the proposed framework to ultimately improve classification accuracy promising experimental results are reported on a toy problem and a paper classification task

confluent kz equations for mathfrak sl _n with poincare rank 2 at infinity we construct confluent kz equations with poincar rank 2 at infinity for the case of and the integral representation for the solutions hamiltonians of these confluent kz equations are derived from suitable quantization of dlog constructed in the theory of monodromy preserving deformation mpd in jimbo et al 1981 physica d 2 306 52 our confluent kz equations can be viewed as a quantization of mpd with poincar rank 2 at infinity

fast approximate k nn graph construction for high dimensional data via recursive lanczos bisection nearest neighbor graphs are widely used in data mining and machine learning a brute force method to compute the exact knn graph takes dn2 time for n data points in the d dimensional euclidean space we propose two divide and conquer methods for computing an approximate knn graph in dnt time for high dimensional data large d the exponent t 1 2 is an increasing function of an internal parameter which governs the size of the common region in the divide step experiments show that a high quality graph can usually be obtained with small overlaps that is for small values of t a few of the practical details of the algorithms are as follows first the divide step uses an inexpensive lanczos procedure to perform recursive spectral bisection after each conquer step an additional refinement step is performed to improve the accuracy of the graph finally a hash table is used to avoid repeating distance calculations during the divide and conquer process the combination of these techniques is shown to yield quite effective algorithms for building knn graphs

a congruence modulo four in real schubert calculus we establish a congruence modulo four in the real schubert calculus on the grassmannian of m planes in 2m space this congruence holds for fibers of the wronski map and a generalization to what we call symmetric schubert problems this strengthens the usual congruence modulo two for numbers of real solutions to geometric problems it also gives examples of geometric problems given by fibers of a map whose degree is zero but where each fiber contains real points

some remarks on real and complex output feedback we provide some new necessary and sufficient conditions which guarantee arbitrary pole placement of a particular linear system over the complex numbers we exhibit a nontrivial real linear system which is not controllable by real static output feedback and discuss a conjecture from algebraic geometry concerning the existence of real linear systems for which all static feedback laws are real

an incremental learning algorithm based on the k associated graph for non stationary data classification non stationary classification problems concern the changes on data distribution over a classifier lifetime to face this problem learning algorithms must conciliate essential but difficult to gather attributes like good classification performance stability and low associated costs like processing time and memory this paper presents an extension of the k associated optimal graph learning algorithm to cope with classification over non stationary domains the algorithm relies on a graph structure consisting of many disconnected components subgraphs such graph enhances data representation by fitting locally groups of data according to a purity measure which in turn quantifies the overlapping between vertices of different classes as a result the graph can be used to accurately estimate the probability of unlabeled data to belong to a given class the proposed algorithm is benefited from the dynamical evolution of the graph by updating its set of components when new data is presented along time by removing old components as new components arise experimental results on artificial and real domains and further statistical analysis show that the proposed algorithm is an effective solution to non stationary classification problems

affine kac moody algebras and semi infinite flag manifolds we study representations of affine kac moody algebras from a geometric point of view it is shown that wakimoto modules introduced in 18 which are important in conformal field theory correspond to certain sheaves on a semi infinite flag manifold with support on its schhubert cells this manifold is equipped with a remarkable semi infinite structure which is discussed in particular the semi infinite homology of this manifold is computed the cousin grothendieck resolution of an invertible sheaf on a semi infinite flag manifold gives a two sided resolution of an irreducible representation of an affine algebras consisting of wakimoto modules this is just the brst complex as a byproduct we compute the homology of an algebra of currents on the real line with values in a nilpotent lie algebra

generalized hypergeometric functions on the torus and the adjoint representation ofu q sl 2 we study the homology groups with coefficient in local systems arising in the free field representation of minimal models of conformal field theory on an elliptic curve with punctures we define an action of the quantum enveloping algebrauq sl2 on a space of relative cycles extending results obtained previously for the sphere absolute cycles are identified with singular vectors in the case of one puncture we find that the resulting topological representation is essentially the adjoint representation

bispectral and gln glm dualities discrete versus differential we define integral transforms which establish a correspondence between spaces of quasi exponentials and spaces of quasi polynomials under certain conditions as a corollary of the properties of our transforms we obtain a correspondence between solutions to the bethe ansatz equations of two gln glm dual quantum integrable models one is the special trigonometric gaudin model and the other is the special xxx model

the evans krylov theorem for nonlocal parabolic fully nonlinear equations abstract in this paper we prove the evans krylov theorem for nonlocal parabolic fully nonlinear equations

local and global mappings of topology representing networks as data analysis tasks often have to deal with complex data structures the nonlinear dimensionality reduction methods play an important role in exploratory data analysis in the literature a number of nonlinear dimensionality reduction techniques have been proposed e g sammon mapping locally linear embedding these techniques attempt to preserve either the local or the global geometry of the original data and they perform metric or non metric dimensionality reduction nevertheless it is difficult to apply most of them to large data sets there is a need for new algorithms that are able to combine vector quantisation and mapping methods in order to visualise the data structure in a low dimensional vector space in this paper we define a new class of algorithms to quantify and disclose the data structure that are based on the topology representing networks and apply different mapping methods to the low dimensional visualisation not only existing methods are combined for that purpose but also a novel group of mapping methods topology representing network map are introduced as a part of this class topology representing network maps utilise the main benefits of the topology representing networks and of the multidimensional scaling methods to disclose the real structure of the data set under study to determine the main properties of the topology representing network based mapping methods a detailed analysis of classical benchmark examples wine and optical recognition of handwritten digits data set is presented

jackson integrals of jordan pochhammer type and quantum knizhnik zamolodchikov equations we show that theq difference systems satisfied by jackson integrals of jordan pochhammer type give a class of the quantum knizhnik zamolodchikov equation for u q widehat mathfrak s mathfrak l 2 in the sense of frenkel and reshetikhin

hypergeometric integrals and arrangements abstract this is an outline of the aomoto gelfand theory of multivariable hypergeometric integrals and varchenko s formula for the determinant of the period matrix of the hypergeometric pairing a significant feature of this work is the use of the theory of arrangements of hyperplanes to transform a problem in analysis into one in combinatorics

error bounds of multi graph regularized semi supervised classification in this paper we investigate the generalization performance of the multi graph regularized semi supervised classification algorithm associated with the hinge loss we provide estimates for the excess misclassification error of multi graph regularized classifiers and show the relations between the generalization performance and the structural invariants of data graphs experiments performed on real database demonstrate the effectiveness of our theoretical analysis

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

quantum algebra structure of certain jackson integrals theq difference system satisfied by jackson integrals with a configuration ofa type root system is studied we explicitly construct some linear combination of jackson integrals which satisfies the quantum knizhnik zamolodchikov equation for the 2 point correlation function ofq vertex operators introduced by frenkel and reshetikhin for the quantum affine algebra u q widehat mathfrak s mathfrak l 2 the expression of integrands for then point case is conjectured and a set of linear relations for the corresponding jackson integrals is proved

an application of aomoto gelfand hypergeometric functions to the su n knizhnik zamolodchikov equation solutions to the knizhnik zamolodchikov equation for verma modules of the lie algebra mathfrak s mathfrak l n 1 mathbb c are explicitly given by certain integrals called aomoto gelfand hypergeometric functions

lower bounds for numbers of real solutions in problems of schubert calculus we give lower bounds for the numbers of real solutions in problems appearing in schubert calculus in the grassmannian mathop rm gr n d related to osculating flags it is known that such solutions are related to bethe vectors in the gaudin model associated to mathop rm gl n the gaudin hamiltonians are self adjoint with respect to a non degenerate indefinite hermitian form our bound comes from the computation of the signature of that form

cordes nirenberg type estimates for nonlocal parabolic equations in this paper we obtain cordes nirenberg type estimates for nonlocal parabolic equations on the more flexible solution space lt l 1 than the classical solution space b rtn consisting of all bounded functions on rtn

galois groups of schubert problems of lines are at least alternating we show that the galois group of any schubert problem involving lines in projective space contains the alternating group this constitutes the largest family of enu merative problems whose galois groups have been largely determined using a criterion of vakil and a special position argument due to schubert our result follows from a particular inequality among kostka numbers of two rowed tableaux in most cases a combinatorial injection proves the inequality for the remaining cases we use the weyl integral formula to obtain an integral formula for these kostka numbers this rewrites the inequality as an integral which we estimate to establish the inequality

quantum knizhnik zamolodchikov equations and affine root systems quantum difference knizhnik zamolodchikov equations s1 fr are generalized for ther matrices from ch1 with the arguments in arbitrary root systems and their formal counterparts in particular qkz equations with certain boundary conditions are introducted the self consistency of the equations from fr and the cross derivative integrability conditions for ther matrix kz equations from ch2 are obtained as corollaries a difference counterpart of the quantum many body problem connected with macdonald s operators is defined as an application

the evans krylov theorem for nonlocal fully nonlinear equations we prove a regularity result for solutions of a purely integro differential bellman equation this regularity is enough for the solutions to be understood in the classical sense if we let the order of the equation approach two we recover the theorem of evans and krylov about the regularity of solutions to concave uniformly elliptic partial differential equations

further time regularity for nonlocal fully nonlinear parabolic equations we establish holder estimates for the time derivative of solutions of nonlocal parabolic equations under mild assumptions for the boundary data as a consequence we are able to extend the evans krylov estimate for rough kernels to parabolic equations 2016 wiley periodicals inc

the b and m shapiro conjecture in real algebraic geometry and the bethe ansatz we prove the b and m shapiro conjecture that if the wronskian of a set of polynomials has real roots only then the complex span of this set of polynomials has a basis consisting of polynomials with real coefficients this in particular implies the following result if all ramification points of a parametrized rational curve 1 r lie on a circle in the riemann sphere 1 then maps this circle into a suitable real subspace r r the proof is based on the bethe ansatz method in the gaudin model the key observation is that a symmetric linear operator on a euclidean space has real spectrum in appendix a we discuss properties of differential operators associated with bethe vectors in the gaudin model in particular we prove a statement which may be useful in complex algebraic geometry it claims that certain schubert cycles in a grassmannian intersect transversally if the spectrum of the corresponding gaudin hamiltonians is simple in appendix b we formulate a conjecture on reality of orbits of critical points of master functions and prove this conjecture for master functions associated with lie algebras of types a r b r and c r

regularity results for fully nonlinear parabolic integro differential operators in this paper we consider the regularity theory for fully nonlinear parabolic integro differential equations with symmetric kernels we are able to find parabolic versions of alexandrov backelman pucci estimate with 0 sigma 2 and we show a harnack inequality holder regularity and c 1 alpha regularity of the solutions by obtaining decay estimates of their level sets

off shell bethe ansatz equations and n point correlators in the su 2 wznw theory we prove that the wavevectors of the off shell bethe ansatz equation for the inhomogeneous su 2 lattice vertex model render in the quasiclassical limit the solutions of the knizhnik zamolodchikov equation

fast k most similar neighbor classifier for mixed data tree k msn the k nearest neighbor k nn classifier has been a widely used nonparametric technique in pattern recognition because of its simplicity and good performance in order to decide the class of a new prototype the k nn classifier performs an exhaustive comparison between the prototype to classify and the prototypes in the training set t however when t is large the exhaustive comparison is expensive for this reason many fast k nn classifiers have been developed some of them are based on a tree structure which is created during a preprocessing phase using the prototypes in t then in a search phase the tree is traversed to find the nearest neighbor the speed up is obtained while the exploration of some parts of the tree is avoided using pruning rules which are usually based on the triangle inequality however in soft sciences as medicine geology sociology etc the prototypes are usually described by numerical and categorical attributes mixed data and sometimes the comparison function for computing the similarity between prototypes does not satisfy metric properties therefore in this work an approximate fast k most similar neighbor classifier for mixed data and similarity functions that do not satisfy metric properties based on a tree structure tree k msn is proposed some experiments with synthetic and real data are presented

a geometric littlewood richardson rule we describe a geometric littlewood richardson rule interpreted as deforming the intersection of two schubert varieties into the union of schubert varieties there are no restrictions on the base eld and all multiplicities arising are 1 this is important for applications this rule should be seen as a generalization of pieri s rule to arbitrary schubert classes by way of explicit homotopies it has straightforward bijections to other littlewood richardson rules such as tableaux and knutson and tao s puzzles this gives the rst geometric proof and interpretation of the littlewood richardson rule geometric consequences are described here and in v2 kv1 kv2 v3 for example the rule also has an interpretation in k theory suggested by buch which gives an extension of puzzles to k theory

properties of graphs preserved by relational graph rewritings abstract we formulate graphs and graph rewritings using binary relations and call them relational graphs and relational graph rewritings in this framework rewriting is defined using a pushout in a category of relational graphs it is known that an important theorem of rewriting systems called critical pair s lemma can be proved using simple and clear categorical properties in this paper we construct treelike graphs and raoult graphs by some relational conditions we give a sufficient condition for rewriting rules and matchings which guarantees the closedness of those graph rewritings these results show that the critical pair s lemma also holds under some conditions for a graph rewriting system in which graphs are restricted to treelike graphs or raoult graphs

integrable connections related to zonal spherical functions we define a system of differential equations of first order for a function valued in the group algebra of the weyl group associated with an arbitrary root system this is equivalent to the system of differential equations given by heckman and opdam which is a deformation of the system satisfied by the zonal spherical function of the riemannian symmetric spaceg k of non compact type when the root system is a n type our equation is related to the knizhnik zamolodchikov equation in conformal field theory

graph based semisupervised learning graph based learning provides a useful approach for modeling data in classification problems in this modeling scenario the relationship between labeled and unlabeled data impacts the construction and performance of classifiers and therefore a semisupervised learning framework is adopted we propose a graph classifier based on kernel smoothing a regularization framework is also introduced and it is shown that the proposed classifier optimizes certain loss functions its performance is assessed on several synthetic and real benchmark data sets with good results especially in settings where only a small fraction of the data are labeled

laplacian eigenmaps for dimensionality reduction and data representation one of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data we consider the problem of constructing a representation for data lying on a low dimensional manifold embedded in a high dimensional space drawing on the correspondence between the graph laplacian the laplace beltrami operator on the manifold and the connections to the heat equation we propose a geometrically motivated algorithm for representing the high dimensional data the algorithm provides a computationally efficient approach to nonlinear dimensionality reduction that has locality preserving properties and a natural connection to clustering some potential applications and illustrative examples are discussed

on algebraic equations satisfied by hypergeometric correlators in wzw models ii the paper contains an explicit description of genus 0 conformal block bundles for wess zumino witten models of conformal field theory we prove that an earlier construction due to the second and the third authors gives a map of these bundles to certain de rham cohomology bundles

graph laplacians and their convergence on random neighborhood graphs given a sample from a probability measure with support on a submanifold in euclidean space one can construct a neighborhood graph which can be seen as an approximation of the submanifold the graph laplacian of such a graph is used in several machine learning methods like semi supervised learning dimensionality reduction and clustering in this paper we determine the pointwise limit of three different graph laplacians used in the literature as the sample size increases and the neighborhood size approaches zero we show that for a uniform measure on the submanifold all graph laplacians have the same limit up to constants however in the case of a non uniform measure on the submanifold only the so called random walk graph laplacian converges to the weighted laplace beltrami operator

a coloring fuzzy graph approach for image classification one of the main problems in practice is the difficulty in dealing with membership functions many decision makers ask for a graphical representation to help them to visualize results in this paper we point out that some useful tools for fuzzy classification can be derived from fuzzy coloring procedures in particular we bring here a crisp grey coloring algorithm based upon a sequential application of a basic black and white binary coloring procedure already introduced in a previous paper d gomez j montero j yanez c poidomani a graph coloring algorithm approach for image segmentation omega in press in this article the image is conceived as a fuzzy graph defined on the set of pixels where fuzzy edges represent the distance between pixels in this way we can obtain a more flexible hierarchical structure of colors which in turn should give useful hints about those classes with unclear boundaries

a history of graph entropy measures this survey seeks to describe methods for measuring the entropy of graphs and to demonstrate the wide applicability of entropy measures setting the scene with a review of classical measures for determining the structural information content of graphs we discuss graph entropy measures which play an important role in a variety of problem areas including biology chemistry and sociology in addition we examine relationships between selected entropy measures illustrating differences quantitatively with concrete examples

boundary quantum knizhnik zamolodchikov equations and bethe vectors solutions to boundary quantum knizhnik zamolodchikov equations are constructed as bilateral sums involving off shell bethe vectors in case the reflection matrix is diagonal and only the 2 dimensional representation of mathcal u q widehat mathfrak sl 2 is involved we also consider their rational and classical degenerations

conformal blocks of minimal models on a riemann surface we give explicit integral representations for conformal blocks of minimal models on arbitrary compact riemann surfaces

hypergeometric solutions of trigonometric kz equations satisfy dynamical difference equations abstract the trigonometric kz equations associated to a lie algebra g depend on a parameter h where h g is a cartan subalgebra a system of dynamical difference equations with respect to compatible with the kz equations is introduced by v tarasov and the second author 2000 internat math res notices 15 801 829 we prove that the standard hypergeometric solutions of the trigonometric kz equations associated to sl n also satisfy the dynamical difference equations

frontiers of reality in schubert calculus the theorem of mukhin tarasov and varchenko formerly the shapiro conjecture for grassmannians asserts that all a priori complex solutions to certain geometric problems in the schubert calculus are actually real their proof is quite remarkable using ideas from integrable systems fuchsian differential equations and representation theory there is now a second proof of this result and it has ramifica tions in other areas of mathematics from curves to control theory to combinatorics despite this work the original shapiro conjecture is not yet settled while it is false as stated it has several interesting and not quite understood modifications and gen eralizations that are likely true and the strongest and most subtle version of the shapiro conjecture for grassmannians remains open

divisors on general curves and cuspidal rational curves in t roduc t ion 371 0 pre l iminar ies 377 1 a pl i icker formula 378 2 d imens iona l t ransversa l i ty 380 3 d i m e n s i o n of families of r a t iona l curves with fixed degree and ass igned cusps in a project ive space 383 4 l inear series on curves with universa l s ingular i t ies 386 5 l inear series and project ive embedd ings of general curves 391 6 embedd ings of general cuspidal curves 400 7 degenera t ion of schuber t in tersect ions some special cases 403 8 degene ra t ion of schuber t in tersect ions ma in results 407 9 g c is reduced for general cuspidal r a t iona l c 414

a new separation measure for improving the effectiveness of validity indices many validity indices have been proposed for quantitatively assessing the performance of clustering algorithms one limitation of existing indices is their lack of generalizability due to their dependence on the specific algorithms and structures of the data space to handle large scale datasets with arbitrary structures this research study proposes a new cluster separation measure for improving the effectiveness of existing validity indices this is achieved by partitioning the original data space into a grid based structure which allows the introduction of a new measurement for assessing the true data distribution between any two clusters instead of the distance between the two cluster prototypes to validate the effectiveness of the proposed separation measure we adopt two commonly used validity indices the davies bouldin s function db and tibshirani s gap statistic gs these indices are denoted as r db 1 and r gs 1 for clusters with sphere shaped structures and r db 2 and r gs 2 for irregular shaped structures this integration enables the indices to evaluate both partitional algorithms and hierarchical algorithms partitional algorithms including c means cm fuzzy c means fcm and hierarchical algorithms including dbscan and clique are used to test the performance of the new indices two synthetic datasets with spherical structures and four synthetic datasets with irregular shapes are first compared five real datasets from the uci machine learning repository are then used to further test the measure s performance the experimental results provide evidence that the new indices outperform the original indices

an analytic approach to purely nonlocal bellman equations arising in models of stochastic control abstract given a bounded domain r d and two integro differential operators l 1 l 2 of the form l j u x p v u x u y k j x y x y d y we study the fully nonlinear bellman equation 0 1 max j 1 2 l j u x a j x u x f j x 0 in with dirichlet boundary conditions here a j f j r are non negative functions we prove the existence of a non negative function u r which satisfies 0 1 almost everywhere the main difficulty arises through the nonlocality of l j and the absence of regularity near the boundary

bosonization and integral representation of solutions of the knizhnik zamolodchikov bernard equations we construct an integral representation of solutions of the knizhnik zamolodchikov bernard equations using the wakimoto modules

rational functions with real critical points and the b and m shapiro conjecture in real enumerative geometry suppose that 2d 2 tangent lines to the rational normal curve z 1 z z d in d dimensional complex projective space are given it was known that the number of codimension 2 subspaces intersecting all these lines is always finite for a generic configuration it is equal to the d th catalan number we prove that for real tangent lines all these codimension 2 subspaces are also real thus confirming a special case of a general conjecture of b and m shapiro this is equivalent to the following result if all critical points of a rational function lie on a circle in the riemann sphere for example on the real line then the function maps this circle into a circle

jeu de taquin and a monodromy problem for wronskians of polynomials abstract the wronskian associates to d linearly independent polynomials of degree at most n a non zero polynomial of degree at most d n d this can be viewed as giving a flat finite morphism from the grassmannian gr d n to projective space of the same dimension in this paper we study the monodromy groupoid of this map when the roots of the wronskian are real we show that the monodromy is combinatorially encoded by schutzenberger s jeu de taquin hence we obtain new geometric interpretations and proofs of a number of results from jeu de taquin theory including the littlewood richardson rule

random fields and the enumerative geometry of lines on real and complex hypersurfaces we introduce a probabilistic framework for the study of real and complex enumerative geometry of lines on hypersurfaces this can be considered as a further step in the original shub smale program of studying the real zeros of random polynomial systems our technique is general and it also applies for example to the case of the enumerative geometry of flats on complete intersections we derive a formula expressing the average number e n of real lines on a random hypersurface of degree 2n 3 in mathbb r text p n in terms of the expected modulus of the determinant of a special random matrix in the case n 3 we prove that the average number of real lines on a random cubic surface in mathbb r text p 3 equals begin aligned e 3 6 sqrt 2 3 end aligned this technique can also be applied to express the number c n of complex lines on a generic hypersurface of degree 2n 3 in mathbb c text p n in terms of the expectation of the square of the modulus of the determinant of a random hermitian matrix as a special case we recover the classical statement c 3 27 we determine at the logarithmic scale the asymptotic of the quantity e n by relating it to c n whose asymptotic has been recently computed in 19 specifically we prove that begin aligned lim n rightarrow infty frac log e n log c n frac 1 2 end aligned finally we show that this approach can be used to compute the number r n 2n 3 of real lines counted with their intrinsic signs as defined in 28 on a generic real hypersurface of degree 2n 3 in mathbb r text p n

a nonparametric classification method based on k associated graphs graph is a powerful representation formalism that has been widely employed in machine learning and data mining in this paper we present a graph based classification method consisting of the construction of a special graph referred to as k associated graph which is capable of representing similarity relationships among data cases and proportion of classes overlapping the main properties of the k associated graphs as well as the classification algorithm are described experimental evaluation indicates that the proposed technique captures topological structure of the training data and leads to good results on classification task particularly for noisy data in comparison to other well known classification techniques the proposed approach shows the following interesting features 1 a new measure called purity is introduced not only to characterize the degree of overlap among classes in the input data set but also to construct the k associated optimal graph for classification 2 nonlinear classification with automatic local adaptation according to the input data contrasting to k nearest neighbor classifier which uses a fixed k the proposed algorithm is able to automatically consider different values of k in order to best fit the corresponding overlap of classes in different data subspaces revealing both the local and global structure of input data 3 the proposed classification algorithm is nonparametric implicating high efficiency and no need for model selection in practical applications