lie theory for nilpotent l algebras the deligne groupoid is a functor from nilpotent differential graded lie algebras concentrated in positive degrees to groupoids in the special case of lie algebras over a field of characteristic zero it gives the associated simply connected lie group we generalize the deligne groupoid to a functor from l algebras concentrated in degree n to n groupoids we actually construct the nerve of the n groupoid which is an enriched kan complex the construction of gamma is quite explicit it is based on dupont s proof of the de rham theorem and yields higher dimensional analogues of holonomy and of the campbell hausdorff formula in the case of abelian l algebras i e chain complexes the functor is the dold kan simplicial set

long time dynamics for semi relativistic nls and half wave in arbitrary dimension we consider the cauchy problems associated with semi relativistic nls snls and half wave hw in particular we focus on the following two main questions local global cauchy theory existence and stability instability of ground states in between other results we prove the existence and stability of ground states for snls in the l 2 supercritical regime this is in sharp contrast with the instability of ground states for the corresponding hw which is also established along the paper by showing an inflation of norms phenomenon concerning the cauchy theory we show under radial symmetry assumption the following results a local existence result in h 1 for energy subcritical nonlinearity and a global existence result in the l 2 subcritical regime

reliable all pairs evolving fuzzy classifiers in this paper we propose a novel design of evolving fuzzy classifiers efcs to handle online multiclass classification problems in a data streaming context therefore we exploit the concept of all pairs ap a k a all versus all classification using binary classifiers for each pair of classes this benefits from less complex decision boundaries to be learned as opposed to a direct multiclass approach and achieves a higher efficiency in terms of incremental training time than one versus rest classification techniques for the binary classifiers we apply fuzzy classifiers with singleton class labels in the consequences as well as takagi sugeno t s fuzzy models to conduct regression on 0 1 for each class pair both are evolved and incrementally trained in a data streaming context yielding a permanent update of the whole ap collection of classifiers thus being able to properly react to dynamic changes in the streams the classification phase considers a novel strategy by using the preference levels of each pair of classes that are collected in a preference relation matrix and performing a weighted voting scheme on this matrix this is done by investigating the reliability of the classifiers in their predictions 1 integrating the degree of ignorance on samples to be classified as weights for the preference levels and 2 new conflict models used in the single binary classifiers and when calculating the final class response based on the preference relation matrix the advantage of the new efc concept over the single model using a direct multiclass classification concept and multimodel architectures using a one versus rest classification concept will be underlined by empirical evaluations and comparisons at the end of this paper based on high dimensional real world multiclass classification problems the results also show that integrating conflict and ignorance concepts into the preference relations can boost classifier accuracies

on a vanishing conjecture appearing in the geometric langlands correspondence 0 1 this paper should be regarded as a sequel to 7 there it was shown that the geometric langlands conjecture for gln follows from a certain vanishing conjecture the goal of the present paper is to prove this vanishing conjecture let x be a smooth projective curve over a ground field k let e be an m dimensional local system on x and let bunm be the moduli stack of rank m vector bundles on x the geometric langlands conjecture says that to e we can associate a perverse sheaf fe on bunm which is a hecke eigensheaf with respect to e the vanishing conjecture of 7 says that for all integers n m a certain functor ave depending on e and a parameter d z which maps the category d bunn to itself vanishes identically when d is large enough the fact that the vanishing conjecture implies the geometric langlands conjecture may be regarded as a geometric version of the converse theorem moreover as will be explained in the sequel the vanishing of the functor ave is analogous to the condition that the rankin selberg convolution of e viewed as an m dimensional galois representation and an automorphic form on gln with n m is well behaved both the geometric langlands conjecture and the vanishing conjecture can be formulated in any of the sheaf theoretic situations e g q adic sheaves when char k d modules when char k 0 and sheaves with coefficients in a finite field f again when char k when the ground field is the finite field fq and we are working with adic coefficients it was shown in 7 that the vanishing conjecture can be deduced from lafforgue s theorem that establishes the full langlands correspondence for global fields of positive characteristic cf 9

evolving fuzzy classifiers using different model architectures in this paper we present two novel approaches for on line evolving fuzzy classifiers called eclass and flexfis class both methods can be applied with different model architectures including single model sm with class labels as consequents classification hyper planes as consequents and multi model mm architecture additionally eclass can have a multi input multi output mimo architecture with multiple hyper planes as consequents of each fuzzy rule the difference between mm and mimo architectures is that the former one applies one separate and independent fuzzy rule based frb classifier for each class and is using an indicator labelling scheme while the latter one applies a single frb where the rules are mimo rather than miso both eclass and flexfis class methods are designed to work on a per sample basis and are thus one pass incremental additionally their structure frb is evolving rather than fixed it adapts their parameters in antecedent and consequent parts with any newly loaded sample a special emphasis is placed on advanced issues for improving accuracy and robustness including a thorough comparison between global and local learning of consequent functions a novel approach for detecting of and reacting on drifts in the data streams and an enhanced outlier treatment strategy the methods and their extensions according to the advanced issues are evaluated on one benchmark problem of handwritten images recognition as well as on a real life problem of image classification framework where images should be classified into good and bad ones during an on line and interactive production process

extensions of vector quantization for incremental clustering in this paper we extend the conventional vector quantization by incorporating a vigilance parameter which steers the tradeoff between plasticity and stability during incremental online learning this is motivated in the adaptive resonance theory art network approach and is exploited in our paper for forming a one pass incremental and evolving variant of vector quantization this variant can be applied for online clustering classification and approximation tasks with an unknown number of clusters additionally two novel extensions are described one concerns the incorporation of the sphere of influence of clusters in the vector quantization learning process by selecting the winning cluster based on the distances of a data point to the surface of all clusters another one introduces a deletion of cluster satellites and an online split and merge strategy clusters are dynamically split and merged after each incremental learning step both strategies prevent the algorithm to generate a wrong cluster partition due to a bad a priori setting of the most essential parameter s the extensions will be applied to clustering of two and high dimensional data within an image classification framework and for model based fault detection based on data driven evolving fuzzy models

sharp upper bound on the blow up rate for the critical nonlinear schrodinger equation we consider the critical nonlinear schrodinger equation iu t delta u u 4 n with initial condition u 0 x u0

on line incremental feature weighting in evolving fuzzy classifiers in this paper we present an approach for addressing the problem of dynamic dimension reduction during on line training evolution and updating of evolving fuzzy classifiers efc the basic idea of our approach is that instead of permanently changing the list of most important features with newly loaded data blocks we generalize the concept of incremental feature selection to an incremental feature weighting approach features are assigned weights in 0 1 according to their importance level these weights are permanently updated during on line mode and guarantee a smooth learning process in the evolving fuzzy classifiers as they change softly and continuously over time in some cases when the weights become approximately 0 an automatic switching off of some features and therefore a soft dimension reduction is achieved two novel incremental feature weighting strategies are proposed in this paper one based on a leave one feature out the other based on a feature wise separability criterion we will describe the integration concept of the feature weights in the evolving fuzzy classifiers using single and multi model architecture where flexfis class sm and flexfis class mm serve as training engines the whole approach of integrated incremental feature weighting in evolving fuzzy classifiers will be evaluated based on high dimensional on line real world classification scenarios and based on data from the internet the results will show that incremental feature weighting in efc in fact helps to reduce curse of dimensionality and therefore guides the evolving fuzzy classifiers to a higher on line predictive power

blow up of h1 solution for the nonlinear schrodinger equation we consider the blow up of the solution in h1 for the following nonlinear schrodinger equation i tu u u p 1u x rn t 0 u 0 x u0 x x rn t 0 where n 2 and 1 4 n p min n 2 n 2 5 we prove that if the initial data u0 in h1 are radially symmetric and have negative energy then the solution of in h1 blows up in finite time we do not assume that xu0 l2 and therefore our result is the generalization of the results of glassey 4 and m tsutsumi 18 for the radially symmetric case

the structure of solutions to the pseudo conformally invariant nonlinear schrodinger equation we study solutions in n of the nonlinear schrodinger equation iu t u u u where is the fixed power 4 n for this particular power these solutions satisfy the pseudo conformal conservation law and the set of solutions is invariant under a related transformation this transformation gives a correspondence between global and non global solutions if

construction of solutions with exactly k blow up points for the schrodinger equation with critical nonlinearity we consider the nonlinear schrodinger equation i partial u mathord left vphantom i partial u partial t right kern nulldelimiterspace partial t delta u left u right 4 mathord left vphantom 4 n right kern nulldelimiterspace n uandu left 0 right varphi left right 1 whereu 0 t n for any given pointsx1 x2 x k in n we construct a solution of eq 1 u t which blows up in a finite timet at exactlyx1 x2 x k in addition we describe the precise behavior of the solutionu t whent t at the blow up points x1 x2 x k and in n x1 x2 x k

on the geometric langlands conjecture let x be a smooth projective curve and g be a reductive group over a finite field fq the field kx of rational functions on x is what number theorists call a global field and we can do the theory of automorphic functions over it namely we consider the quotient g kx g a here a is the ring of adeles corresponding to kx and study the space of functions on it as a representation of the adele group g a it was essentially an observation of a weil that the quotient g kx g a is closely related to the set of isomorphism classes of principal g bundles on our curve x however g bundles on x possess a richer structure one can view the above set as a set of fq points of an algebraic variety or rather a stack denoted bung moreover instead of studying the vector space of functions on the discrete set of points of bung we will consider the category of adic sheaves on it basically what people call the geometric langlands program is an attempt to understand a spectral decomposition of the above category under the action of the so called hecke functors the latter will be defined in the talk the answer predicted by the geometric langlands conjecture links this spectral decomposition to the moduli stack of local systems on x with respect to the langlands dual group

flexfis a robust incremental learning approach for evolving takagi sugeno fuzzy models in this paper we introduce a new algorithm for incremental learning of a specific form of takagi sugeno fuzzy systems proposed by wang and mendel in 1992 the new data driven online learning approach includes not only the adaptation of linear parameters appearing in the rule consequents but also the incremental learning of premise parameters appearing in the membership functions fuzzy sets together with a rule learning strategy in sample mode a modified version of vector quantization is exploited for rule evolution and an incremental learning of the rules premise parts the modifications include an automatic generation of new clusters based on the nature distribution and quality of new data and an alternative strategy for selecting the winning cluster rule in each incremental learning step antecedent and consequent learning are connected in a stable manner meaning that a convergence toward the optimal parameter set in the least squares sense can be achieved an evaluation and a comparison to conventional batch methods based on static and dynamic process models are presented for high dimensional data recorded at engine test benches and at rolling mills for the latter the obtained data driven fuzzy models are even compared with an analytical physical model furthermore a comparison with other evolving fuzzy systems approaches is carried out based on nonlinear dynamic system identification tasks and a three input nonlinear function approximation example

nonlinear schrodinger equations and sharp interpolation estimates a sharp sufficient condition for global existence is obtained for the nonlinear schrodinger equation begin array 20 c nls 2i phi t delta phi left phi right 2 sigma phi 0 x in mathbb r n t in mathbb r end array in the case 2 n this condition is in terms of an exact stationary solution nonlinear ground state of nls it is derived by solving a variational problem to obtain the best constant for classical interpolation estimates of nirenberg and gagliardo

chern simons theory with finite gauge group we construct in detail a 2 1 dimensional gauge field theory with finite gauge group in this case the path integral reduces to a finite sum so there are no analytic problems with the quantization the theory was originally introduced by dijkgraaf and witten without details the point of working it out carefully is to focus on the algebraic structure and particularly the construction of quantum hilbert spaces on closed surfaces by cutting and pasting this includes the verlinde formula the careful development may serve as a model for dealing with similar issues in more complicated cases

a remark on the blowing up of solutions to the cauchy problem for nonlinear schrodinger equations on considere des solutions de iu t u u p 1 u u o u 0 ou x appartient a un domaine lisse cr n et on demontre que sous de bonnes conditions sur p n et u 0 h 2 h 0 1 u t l 2 eclate en un temps fini

l2 concentration of blow up solutions for the nonlinear schrodinger equation with critical power nonlinearity abstract we consider the solution of the nonlinear schrodinger equation i u t u f u and u 0 where u is defined on 0 t rn and f is a nonlinear complex valued function we consider the case of the critical power that is f z z 4 n z as z in a sense which will be made precise in addition we suppose that u t blows up in the norm h1 at time t we prove that u t has no limit in l2 as t t in the particular case where has a spherical symmetry we further show a phenomenon of l2 concentration at the origin

unstable ground state of nonlinear klein gordon equations on demontre l instabilite de l etat fondamental des equations de klein gordon non lineaires a n dimensions d espace n 3

global existence and compact attractors for the discrete nonlinear schrodinger equation abstract we study the asymptotic behavior of solutions of discrete nonlinear schrodinger type dnls equations for a conservative system we consider the global in time solvability and the question of existence of standing wave solutions similarities and differences with the continuous counterpart nls partial differential equation are pointed out for a dissipative system we prove existence of a global attractor and its stability under finite dimensional approximations similar questions are treated in a weighted phase space finally we propose possible extensions for various types of dnls equations

blow up results of viriel type for zakharov equations we consider the zakharov equations in n for n 2 n 3 we first establish a viriel identity for such equations and then prove a blow up result for solutions with a negative energy

higher algebraic structures and quantization we derive quasi quantum groups in 2 1 dimensional topological field theory directly from the classical action and the path integral detailed computations are carried out for the chern simons theory with finite gauge group the principles behind our computations are presumably more general we extend the classical action in ad 1 dimensional topological theory to manifolds of dimension less thand 1 we then construct a generalized path integral which ind 1 dimensions reduces to the standard one and ind dimensions reproduces the quantum hilbert space in a 2 1 dimensional topological theory the path integral over the circle is the category of representations of a quasi quantum group in this paper we only consider finite theories in which the generalized path integral reduces to a finite sum new ideas are needed to extend beyond the finite theories treated here

geometric langlands duality and representations of algebraic groups over commutative rings as such it can be viewed as a first step in the geometric langlands program the connected complex reductive groups have a combinatorial classification by their root data in the root datum the roots and the co roots appear in a symmetric manner and so the connected reductive algebraic groups come in pairs if g is a reductive group we write g for its companion and call it the dual group g the notion of the dual group itself does not appear in satake s paper but was introduced by langlands together with its various elaborations in li l2 and is a cornerstone of the langlands program it also appeared later in physics mo gno in this paper we discuss the basic relationship between g and g we begin with a reductive g and consider the affine grassmannian qx the grassmannian for the loop group of g for technical reasons we work with formal algebraic loops the affine grassmannian is an infinite dimen sional complex space we consider a certain category of sheaves the spherical perverse sheaves on r these sheaves can be multiplied using a convolution product and this leads to a rather explicit construction of a hopf algebra by what has come to be known as tannakian formalism the resulting hopf algebra turns out to be the ring of functions on g in this interpretation the spherical perverse sheaves on the affine grassman nian correspond to finite dimensional complex representations of g thus instead of defining g in terms of the classification of reductive groups we pro vide a canonical construction of g starting from g we can carry out our construction over the integers the spherical perverse sheaves are then those with integral coefficients but the grassmannian remains a complex algebraic object

limit of the solution of a nonlinear schrodinger equation at blow up time we consider the solution of the equation i v t and 0 where v is defined on 0 t rn and is a nonlinear complex valued function let us denote the function x v t x by v t we suppose that v blows up in the norm h1 at time t under suitable assumptions on and we prove the existence of a limit for v t when t t in appropriate spaces furthermore we specify the behavior of v near blow up time

geometric langlands and the equations of nahm and bogomolny geometric langlands duality relates a representation of a simple lie group g vee to the cohomology of a certain moduli space associated with the dual group g in this correspondence a principal sl 2 subgroup of g vee makes an unexpected appearance why this happens can be explained using gauge theory as we will see in this article with the help of the equations of nahm and bogomolny based on a lecture at geometry and physics atiyah 80 edinburgh april 2009

associating visual textures with human perceptions using genetic algorithms this paper deals with an approach allowing to associate visual textures with given human perceptions hereby based on a forward model associating human perceptions for given visual textures the deduction of an reverse process is presented which is able to associate and characterize visual textures for given human perceptions for doing so we propose a constraint based genetic algorithm approach which is able to minimize a specific optimization problem containing constraints in form of band widths for valid individuals low level features extracted from textures in a population the constraints are determined by relationships between low level features characterizing textures in form of high dimensional approximation models additionally in each iteration step checking for valid individuals is carried out with a texture non texture classifier or by using a convex hull over a set of valid textures the whole approach is evaluated based on a real world texture set used as a start population in the genetic algorithm and by defining various kinds of human perceptions for which textures are sought represented by adjective vectors in the aesthetic space the generated individuals low level feature vectors have a high level of fitness they are quite close to the pre defined adjective vectors and a small distance to the initial population the textures synthesized based on the generated individuals are visualized and compared with textures synthesized by a time intensive direct texture mixing and re combination method based on a real world texture data base

neural gas network for vector quantization and its application to time series prediction a neural network algorithm based on a soft max adaptation rule is presented this algorithm exhibits good performance in reaching the optimum minimization of a cost function for vector quantization data compression the soft max rule employed is an extension of the standard k means clustering procedure and takes into account a neighborhood ranking of the reference weight vectors it is shown that the dynamics of the reference weight vectors during the input driven adaptation procedure are determined by the gradient of an energy function whose shape can be modulated through a neighborhood determining parameter and resemble the dynamics of brownian particles moving in a potential determined by the data point density the network is used to represent the attractor of the mackey glass equation and to predict the mackey glass time series with additional local linear mappings for generating output values the results obtained for the time series prediction compare favorably with the results achieved by backpropagation and radial basis function networks

multivariable gaussian evolving fuzzy modeling system this paper introduces a class of evolving fuzzy rule based system as an approach for multivariable gaussian adaptive fuzzy modeling the system is an evolving takagi sugeno ets functional fuzzy model whose rule base can be continuously updated using a new recursive clustering algorithm based on participatory learning the fuzzy sets of the rule antecedents are multivariable gaussian membership functions which have been adopted to preserve information between input variable interactions the parameters of the membership functions are estimated by the clustering algorithm a weighted recursive least squares algorithm updates the parameters of the rule consequents experiments considering time series forecasting and nonlinear system identification are performed to evaluate the performance of the approach proposed the multivariable gaussian evolving fuzzy models are compared with alternative evolving fuzzy models and classic models with fixed structures the results suggest that multivariable gaussian evolving fuzzy modeling is a promising approach for adaptive system modeling

higher dimensional algebra and topological quantum field theory the study of topological quantum field theories increasingly relies upon concepts from higher dimensional algebra such as n categories and n vector spaces we review progress towards a definition of n category suited for this purpose and outline a program in which n dimensional topological quantum field theories tqfts are to be described as n category representations first we describe a suspension operation on n categories and hypothesize that the k fold suspension of a weak n category stabilizes for k n 2 we give evidence for this hypothesis and describe its relation to stable homotopy theory we then propose a description of n dimensional unitary extended tqfts as weak n functors from the free stable weak n category with duals on one object to the n category of n hilbert spaces we conclude by describing n categorical generalizations of deformation quantization and the quantum double construction

a physical origin for singular support conditions in geometric langlands theory we explain how the nilpotent singular support condition introduced into the geometric langlands conjecture by arinkin and gaitsgory arises naturally from the point of view of 4 dimensional mathcal n 4 supersymmetric gauge theory we define what it means in topological quantum field theory to restrict a category of boundary conditions to the full subcategory of objects compatible with a fixed choice of vacuum both in functorial field theory and in the language of factorization algebras for b twisted mathcal n 4 gauge theory with gauge group g the moduli space of vacua is equivalent to mathfrak h w and the nilpotent singular support condition arises by restricting to the vacuum 0 in mathfrak h w we then investigate the categories obtained by restricting to points in larger strata and conjecture that these categories are equivalent to the geometric langlands categories with gauge symmetry broken to a levi subgroup and furthermore that by assembling such for the groups mathrm gl n with n ge 1 one finds a hidden factorization algebra structure for the geometric langlands theory

fitting an unknown number of lines and planes to image data through compatible cluster merging abstract a compatible cluster merging algorithm is presented that is specially designed to find the optimum number of linear planar and hyperplanar clusters i e clusters that lie in a subspace of the original space when an upper bound on the number of clusters present is known this algorithm is shown to be superior to more traditional validity measure based approaches the effectiveness and advantages of the proposed technique in 2d and 3d applications is demonstrated with both synthetic and real data the proposed applications include character recognition obtaining straight line descriptions of intensity edge images and obtaining planar approximations of 3d range data

d branes categories and n 1 supersymmetry we show that boundary conditions in topological open string theory on calabi yau cy manifolds are objects in the derived category of coherent sheaves as foreseen in the homological mirror symmetry proposal of kontsevich together with conformal field theory considerations this leads to a precise criterion determining the supersymmetry preserving branes at any point in cy moduli space completing the proposal of ii stability

joint block structure sparse representation for multi input multi output mimo t s fuzzy system identification takagi sugeno t s fuzzy system identification provides a reasonable framework for modeling and approximation by decomposition of a nonlinear system into a collection of local linear models in this paper we extend the multidimensional output fuzzy rule from the corresponding single output fuzzy rule with the important difference in the variable definition of output namely the output of each fuzzy rule is represented by a multidimensional vector instead of a scalar value in this case the consequent of the fuzzy rule with the multidimensional output variable share a common antecedent part different from the traditional methods that separate the multi input multi output fuzzy system into a group of multi input single output fuzzy models in this paper we take into account the block structure information in the t s fuzzy system and cast the problem of multidimensional output fuzzy model identification as a joint structure sparse optimization problem where the consequent parameters are estimated with a common block structured sparsity pattern over all dimensions of the output variable furthermore we exploit a joint block sparse orthogonal match pursuit algorithm to reduce the number of fuzzy rules in terms of all dimensions of the output variable and prove the sufficient conditions in consideration of the multidimensional output together with the block structure in the t s fuzzy model this method is efficient and shows good performance in well known benchmark datasets and real world problems