on resolving the multiplicity of arbitrary tensor products of the u n groups representations of u n are realised as right translations on holomorphic hilbert bargmann spaces of n n complex variables r fold tensor product spaces of irreducible representations of u n are shown to be isomorphic to subspaces of the holomorphic hilbert spaces maps are exhibited which carry an irreducible representation of u n into these subspaces the algebra of operators commuting with these maps is constructed and it is shown how eigenvalues of certain of these operators can be used to resolve the multiplicity several examples from u 3 are explicitly worked out

on resolving the multiplicity of the branching rule gl 2k 1 c down arrow so 2k 1 c we consider the multiplicity problem of the branching rule gl 2k 1 c down arrow so 2k 1 c finite dimensional irreducible representations of gl 2k 1 c are realized as right translations on subspaces of the holomorphic hilbert bargmann spaces of q 2k 1 complex variables maps are exhibited which carry an irreducible representation of so 2k 1 c into these subspaces an algebra of commuting operators is constructed eigenvalues and eigenvectors of certain of these operators can then be used to resolve the multiplicity in the branching rule

fuzzy clustering analysis for optimizing fuzzy membership functions fuzzy model identification is an application of fuzzy inference system for identifying unknown functions for a given set of sampled data the most important thing for fuzzy identification task is to decide the parameters of membership functions mfs used in fuzzy systems a lot of efforts chung and lee 1994 jang 1993 sun and jang 1993 have been given to initialize the parameters of fuzzy membership functions however the problems of parameter identification were not solved formally assessments of these algorithms are discussed in the paper based on the fuzzy c means fcm bezdek 1987 clustering algorithm we propose a heuristic method to calibrate the fuzzy exponent iteratively a hybrid learning algorithm for refining the system parameters is then presented examples are demonstrated to show the effectiveness of the proposed method comparing with the equalized universe method eum and subtractive clustering method scm chiu 1994 the simulation results indicate the general applicability of our methods to a wide range of applications

data driven fuzzy modeling for takagi sugeno kang fuzzy system this paper presents a systematic approach to design first order tagaki sugeno kang tsk fuzzy systems this approach attempts to obtain the fuzzy rules without any assumption about the structure of the data the structure identification and parameter optimization steps in this approach are carried out automatically and are capable of finding the optimal number of the rules with an acceptable accuracy starting with an initial structure the system first tries to improve the structure and then as soon as an improved structure is found it fine tunes its rules parameters then it goes back to improve the structure again to find a better structure and re fine tune the rules parameters this loop continues until a satisfactory solution tsk model is found the proposed approach has successfully been applied to well known benchmark datasets and real world problems the obtained results are compared with those obtained with other methods from the literature experimental studies demonstrate that the predicted properties have a good agreement with the measured data by using the elicited fuzzy model with a small number of rules finally as a case study the proposed approach is applied to the desulfurization process of a real steel industry comparing the proposed approach with some other fuzzy systems and neural networks it is shown that the developed tsk fuzzy system exhibits better results with higher accuracy and smaller size of architecture

successive identification of a fuzzy model and its applications to prediction of a complex system abstract a successive identification method of a fuzzy model is suggested the identification mechanism consists of two levels one is the supervisor level and the other is the adjustment level the supervisor level determines a policy of parameter adjustment using a set of fuzzy adjustment rules the adjustment rules are derived from the fuzzy implications of a fuzzy model and are extended to fuzzy adjustment rules by using an extended concept of zadeh s contrast intensification the adjustment level executes the policy of parameter adjustment determined with the fuzzy adjustment rules the parameter adjustment consists of premise parameter adjustment and consequent parameter adjustment both of them are realized by the weighted recursive least square algorithm finally it is shown from two examples that the method is very useful for modeling complex systems

on resolving the multiplicity of tensor products of irreducible representations of symplectic groups the multiplicity problem of r fold tensor products of irreducible representations of sp 2k c are considered the arbitrary r fold tensor product is shown to be isomorphic to a subspace of the holomorphic hilbert bargmann space of n 2k complex variables maps are constructed which carry an irreducible representation of sp 2k c into this subspace an algebra of commuting operators is constructed and it is shown how eigenvalues and eigenvectors of certain of these operators can be used to resolve the multiplicity

shift operators and the u n multiplicity problem a computationally effective method for decomposing r fold tensor products of irreducible representations of u n in a basis independent fashion is given the multiplicity arising from the tensor decomposition is resolved with the eigenvalues of invariant operators chosen from the universal enveloping algebra generated by the infinitesimal operators of the dual or complementary representation shift operators which commute with the u n invariant operators but not the dual invariant operators are introduced to compute the eigenvectors and eigenvalues of the dual invariant operators algebraically a three fold tensor product of irreducible representations of su 4 is decomposed to illustrate the power and generality of the method

on non commutative g2 structure using an algebraic orbifold method we present non commutative aspects of g2 structure of seven dimensional real manifolds we first develop and solve the non commutativity parameter constraint equations defining g2 manifold algebras we show that there are eight possible solutions for this extended structure one of which corresponds to the commutative case then we obtain a matrix representation solving such algebras using combinatorial arguments an application to matrix model of m theory is discussed

multiplicity invariants and tensor product decompositions of compact groups decomposing tensor products of irreducible representations of compact groups almost always involves multiplicity wherein some irreducible representations occur more than once in the direct sum decomposition we show that the multiplicity can always be specified by polynomial group invariants the setting is a bargmann segal fock space in n n complex variables where n is the number of labels needed to specify the tensor product and n is the dimension of the fundamental representation of the compact group both the tensor product and direct sum bases are realized as polynomials in this space and it is shown how clebsch gordan and racah coefficients can be computed by suitably differentiating these polynomials the example of su n is discussed in detail and it is shown that the multiplicity can be computed as the solution of certain diophantine equations arising from powers of group invariants namely minors of determinants

a hierarchical fuzzy clustering approach to fuzzy modeling abstract this paper introduces a new method for fuzzy modeling based on a hierarchical fuzzy clustering scheme the method consists of a sequence of steps aiming towards developing a takagi sugeno ts fuzzy model of optimal structure where the fuzzy sets in the premise part are of gaussian type starting from an initial ordinary fuzzy partition of the input space the algorithm performs a nearest neighbor search and groups the original input training data into a number of clusters the centers of these clusters are further processed using an optimal fuzzy clustering technique which is based on the weighted fuzzy c means algorithm the resulted optimal fuzzy partition defines the number of fuzzy rules and provides an initial estimation for the system parameters which in a next step are fine tuned using the well known gradient descend algorithm the proposed method is successfully applied to three test examples where the produced fuzzy models prove to be very accurate as well as compact in size

self tuning fuzzy modeling with adaptive membership function rules and hierarchical structure based on genetic algorithm abstract recently fuzzy systems have been used in many fields and places in order to apply the fuzzy system to the various fields the tuning and optimizing method of the fuzzy system is the key issue some self tuning methods have been proposed so far however these conventional self tuning methods do not have sufficient capability of learning in this paper we propose a new supervised self tuning fuzzy modeling which consist of some membership function expressed by the radial basis function with insensitive region learning is carried out by the genetic algorithms the descent method is also utilized for tuning the shapes of the membership function and consequent parts the effectiveness of the proposed methods is shown by some numerical examples

on noncommutative orbifolds of k3 surfaces using the algebraic geometry method of berenstein and leigh for the construction of the toroidal orbifold t2 t2 t2 z2 z2 with discrete torsion and considering local k3 surfaces we present noncommutative aspects of the orbifolds of product of k3 surfaces in this way the ordinary complex deformation of k3 can be identified with the resolution of stringy singularities by noncommutative algebras using crossed products we give representations and make some comments regarding the fractionation of branes illustrating examples are presented

new perspective on the u n wigner racah calculus ii elementary reduced wigner coefficients for u n for pt i see ibid vol 20 no 9 p 2241 50 1987 exploiting the powerful formalism of the vector coherent state theory expounded in part i the group theoretical content of the complementarity principle and a novel interpretation of the operator pattern of biedenharn and louck the authors rederive in a simple fashion all u n u n 1 reduced wigner coefficients for the elementary u n tensors

a resolution of the su 3 outer multiplicity problem and computation of wigner coefficients for su 3 a simple algorithm is given for the resolution of the su 3 multiplicity problem and the computation of su 3 wigner coefficients using a complete set of u 2 x u 3 bargmann tensors classified by operator patterns null space properties of these tensors are easily derived their structure is such that a direct one to one correspondence is shown to exist between the bargmann tensors and the terms in the clebsch gordan series for su 3 as derived by o reilly 1982 finally the resolution presented herein is shown to be concordant with an alternative resolution advocated long ago by hecht

mirror manifolds in higher dimension we describe mirror manifolds in dimensions different from the familiar case of complex threefolds we isolate certain simplifying features present only in dimension three and supply alternative methods that do not rely on these special characteristics and hence can be generalized to other dimensions although the moduli spaces for calabi yaud folds are not special kahler manifolds whend 3 they still have a restricted geometry and we indicate the new geometrical structures which arise we formulate and apply procedures which allow for the construction of mirror maps and the calculation of order by order instanton corrections to yukawa couplings mathematically these corrections are expected to correspond to calculating chern classes of various parameter spaces hilbert schemes for rational curves on calabi yau manifolds our mirror aided calculations agree with those chern class calculations in the limited number of cases for which the latter can be carried out with current mathematical tools finally we make explicit some striking relations between instanton corrections for various yukawa couplings derived from the associativity of the operator product algebra

a neuro fuzzy system modeling with self constructing rule generationand hybrid svd based learning we propose an approach for neuro fuzzy system modeling a neuro fuzzy system for a given set of input output data is obtained in two steps first the data set is partitioned automatically into a set of clusters based on input similarity and output similarity tests membership functions associated with each cluster are defined according to statistical means and variances of the data points included in the cluster then a fuzzy if then rule is extracted from each cluster to form a fuzzy rule base second a fuzzy neural network is constructed accordingly and parameters are refined to increase the precision of the fuzzy rule base to decrease the size of the search space and to speed up the convergence we develop a hybrid learning algorithm which combines a recursive singular value decomposition based least squares estimator and the gradient descent method the proposed approach has advantages of determining the number of rules automatically and matching membership functions closely with the real distribution of the training data points besides it learns faster consumes less memory and produces lower approximation errors than other methods

canonical orthonormal basis for su 3 so 3 ii reduced matrix elements of the su 3 generators for pt i see ibid vol 18 p 1891 1985 a simple algorithm is given for the calculation of reduced matrix elements of the generators of the su 3 algebra in the canonical su 3 contains implies so 3 basis introduced in i

identification of fuzzy models using a successive tuning method with a variant identification ratio the paper is concerned with the identification of fuzzy inference systems fuzzy models realized with the aid of the successive tuning method using a variant identification ratio that is based on hierarchical fair competition based parallel genetic algorithms hfc pga and information granulation the hfc pga is a certain multi population version of parallel genetic algorithms pga which is suitable for a simultaneous optimization of both the structure of the fuzzy model as well as its parameters the granulation of information is realized with the aid of the c means clustering algorithm information granules formed in this way become essential at further stages of the construction of the fuzzy models by forming the centers modal values of the fuzzy sets constituting individual rules of the inference schemes further optimization of the fuzzy model deals with an adjustment of a suite of parameters such as the number of input variables to be used in the model a collection of specific subsets of the input variables and the number of membership functions being used by these variables and the order and parameters of the polynomial occurring in the conclusions of the corresponding rules an iterative development of the fuzzy model deals with its structural as well as parametric optimization via hfc pga the c means algorithm and a standard least square estimation method to evaluate the performance of the proposed model we exploit well known and commonly used data sets such as gas furnace mackey glass time series as well as medical imaging system a comparative analysis demonstrates that the proposed model leads to the superior performance when compared with other fuzzy models reported in the literature

invariant theory of the block diagonal subgroups of gl n c and generalized casimir operators abstract let c be the field of complex numbers and let gl n c denote the general linear group of order n over c let n be the sum of m positive integers p 1 p m and consider the block diagonal subgroup gl p 1 c x x gl p m c the adjoint representation of the lie group gl n c on its lie algebra gives rise to the coadjoint representation of gl n c on the symmetric algebra of all polynomial functions on gl n c the polynomials that are fixed by the restriction of the coadjoint representation to the block diagonal subgroup form a subalgebra called the algebra of invariants a finite set of generators of this algebra is explicitly determined and the connection with the generalized casimir invariant differential operators is established

structure identification and parameter optimization for non linear fuzzy modeling this work presents a method for non linear fuzzy model identification the main characteristic of the method is the automatic determination of the number and position of the fuzzy sets in the domain of each variable the resultant fuzzy rule base allows model interpretation by domain experts the main contribution of this work is a formulation that allows the optimization of output parameters by a least squares error lse minimization a numerical solution of the lse problem is developed based on the singular value decomposition of the regressor matrix the whole methodology is applied to some numerical examples found in the literature

on the structure of tensor operators in su3 a global algebraic formulation of su3 tensor operator structure is achieved a single irreducible unitary representation irrep v ofso 6 2 is constructed which contains every su3 irrep precisely once an algebra of polynomial differential operatorsa acting onv is given the algebraa is shown to consist of linear combinations of all su3 tensor operators with polynomial invariant operators as coefficients by carrying out an analysis ofa the multiplicity problem for su3 tensor operators is resolved

f theory duals of m theory on g2 manifolds from mirror symmetry using mirror pairs m3 w3 in type ii superstring compactifications on calabi yau threefolds we study geometrically f theory duals of m theory on seven manifolds with g2 holonomy we first develop a way of obtaining landau ginzburg lg calabi yau threefolds w3 embedded in four complex dimensional toric varieties mirror to the sigma model on toric calabi yau threefolds m3 this method gives directly the right dimension without introducing non dynamical variables then using toric geometry tools we discuss the duality between m theory on s1 m3 z2 with g2 holonomy and f theory on elliptically fibred calabi yau fourfolds with su 4 holonomy containing w3 mirror manifolds illustrative examples are presented

calculation of clebsch gordan and racah coefficients using symbolic manipulatio n programs a procedure for calculating clebsch gordan and racah coefficients arising from the decomposition of n fold tensor products of the u n groups using symbolic manipulation programs is given basic states are realized as polynomials in a space equipped with a differentiation inner product the desired coefficients are then obtained by differentiating the relevant polynomials examples from su 2 and su 3 are given copyright 1989 academic press inc

an identification algorithm in fuzzy relational systems abstract the paper presents an effective identification method in fuzzy relational systems we propose an algorithm for constructing models on the basis of fuzzy and nonfuzzy data with the aid of fuzzy discretization and clustering techniques the usefulness of the method provided is demonstrated by means of two numerical examples also a possible way of generating a linguistic decision making algorithm is discussed

invariant theory generalized casimir operators and tensor product decompositions of u n one of the central problems in the representation theory of compact groups concerns multiplicity wherein an irreducible representation occurs more than once in the decomposition of the n fold tensor product of irreducible representations the problem is that there are no operators arising from the group itself whose eigenvalues can be used to label the equivalent representations occurring in the decomposition in this paper we use invariant theory along with so called generalized casimir operators to show how to resolve the multiplicity problem for the u n groups the starting point is to augment the n fold tensor product space with the contragredient representation of interest and construct a subspace of u n invariants the setting for this construction is a polynomial space embedded in a fock space of complex variables which carries all the irreducible representations of u n or gln the dimension of the invariant subspace is equal to the multiplicity occurring in the tensor product decomposition generalized casimir operators are operators from the universal enveloping algebra of outer product u n groups that commute with the diagonal u n action and whose eigenvalues can be used to label the multiplicity using the notion of dual representations we show how to rewrite the generalized casimir operators and prove that they act invariantly on the invariant subspace a complete set of commuting generalized casimir operators can therefore be used to construct eigenvectors that form an orthonormal basis in the invariant subspace different sets of generalized commuting casimir operators generate different orthonormal bases in the invariant subspace the overlaps between the eigenvectors of different commuting sets of generalized casimir operators are called invariant coefficients we show that racah coefficients are special cases of invariant coefficients in which the generalized casimir operators have been chosen with respect to a definite coupling scheme in the tensor product the paper concludes with an example of the threefold tensor product of the eight dimensional irreducible representation of u 3 in which the multiplicity of the chosen irreducible representation is 6 eigenvectors in the six dimensional invariant subspace are computed for different sets of generalized casimir operators and invariant coefficients including racah coefficients

discrete interval type 2 fuzzy system models using uncertainty in learning parameters fuzzy system modeling fsm is one of the most prominent tools that can be used to identify the behavior of highly nonlinear systems with uncertainty conventional fsm techniques utilize type 1 fuzzy sets in order to capture the uncertainty in the system however since type 1 fuzzy sets express the belongingness of a crisp value x of a base variable x in a fuzzy set a by a crisp membership value mua x they cannot fully capture the uncertainties due to imprecision in identifying membership functions higher types of fuzzy sets can be a remedy to address this issue since the computational complexity of operations on fuzzy sets are increasing with the increasing type of the fuzzy set the use of type 2 fuzzy sets and linguistic logical connectives drew a considerable amount of attention in the realm of fuzzy system modeling in the last two decades in this paper we propose a black box methodology that can identify robust type 2 takagi sugeno mizumoto and linguistic fuzzy system models with high predictive power one of the essential problems of type 2 fuzzy system models is computational complexity in order to remedy this problem discrete interval valued type 2 fuzzy system models are proposed with type reduction in the proposed fuzzy system modeling methods fuzzy c means fcm clustering algorithm is used in order to identify the system structure the proposed discrete interval valued type 2 fuzzy system models are generated by a learning parameter of fcm known as the level of membership and its variation over a specific set of values which generate the uncertainty associated with the system structure

mirror symmetry and landau ginzburg calabi yau superpotentials in f theory compactifications we study the landau ginzburg lg theory mirror to two dimensional 2d n 2 gauged linear sigma models on toric calabi yau manifolds we derive and solve new constraint equations for lg elliptic calabi yau superpotentials depending on the physical data of dual linear sigma models in the calabi yau threefold case we consider two examples first we give the mirror symmetry of the canonical line bundle over the hirzebruch surfaces fn second we find a special geometry with the affine so 8 lie algebra toric data extending the geometry of elliptically fibred k3 this geometry leads to a pure n 1 six dimensional so 8 gauge model from the f theory compactification for calabi yau fourfolds we give a new algebraic realization for ade hypersurfaces

quantum screening operators and canonical q de rham cocycles in this paper a close connection is established between certain cohomology spaces of representations of the quantum affine algebra and a twisted q de rham jackson aomoto cohomology of configuration spaces using the quantum screening operators

nc calabi yau orbifolds in toric varieties with discrete torsion using the algebraic geometric approach of berenstein et al hep th 005087 and hep th 009209 and methods of toric geometry we study non commutative nc orbifolds of calabi yau hypersurfaces in toric varieties with discrete torsion we first develop a new way of getting complex d mirror calabi yau hypersurfaces h d in toric manifolds m d 1 with a c r action and analyse the general group of the discrete isometries of h d then we build a general class of d complex dimensional nc mirror calabi yau orbifolds where the non commutativity parameters are solved in terms of discrete torsion and toric geometry data of m d 1 in which the original calabi yau hypersurfaces are embedded next we work out a generalization of the nc algebra for generic d dimensional nc calabi yau manifolds and give various representations depending on different choices of the calabi yau toric geometry data we also study fractional d branes at orbifold points we refine and extend the result for nc t 2 x t 2 x t 2 z 2 x z 2 to higher dimensional torii orbifolds in terms of clifford algebra

clebsch gordan coefficients of su 3 in su 2 and so 3 bases new algorithms are developed for the purpose of optimizing the efficient calculation of su 3 clebsch gordan coefficients in both su 2 and so 3 coupled bases the new algorithms make use of the fact that highest weight states in a tensor product space are easily identified by vector coherent state methods the methods are developed for su 3 but apply to other compact semi simple lie groups

a genetic learning process for the scaling factors granularity and contexts of the fuzzy rule based system data base abstract in this contribution we propose a new method to automatically learn the knowledge base of a fuzzy rule based system by finding an appropriate data base using a genetic algorithm and considering a simple generation method to derive the rule base our genetic process learns all the components of the data base number of labels working ranges and membership function shapes for each linguistic variable using a non linear scaling function to adapt the fuzzy partition contexts

identification of fuzzy systems by means of an auto tuning algorithm and its application to nonlinear systems abstract the study concerns a design procedure of rule based systems the proposed rule based fuzzy modeling implements system structure and parameter identification in the efficient form of if then statements and exploits the theory of system optimization and fuzzy implication rules two types of methods for rule based fuzzy modeling are studied this classification concerns the form of the conclusion part of the rules that can be either constant or formed by some linear functions both triangular and gaussian like membership function are studied the optimization hinges on an auto tuning algorithm that covers a modified constrained optimization method known as a complex method the study introduces a weighted performance index objective function that helps achieve a sound balance between the quality of results produced for the training and testing set this methodology sheds light on the role and impact of different parameters of the model on its performance in particular the mapping and predicting capabilities of the rule based computing the study is illustrated with the aid of several representative numerical examples

genetic programming for model selection of tsk fuzzy systems this paper compares a genetic programming gp approach with a greedy partition algorithm lolimot for structure identification of local linear neuro fuzzy models the crisp linear conclusion part of a takagi sugeno kang tsk fuzzy rule describes the underlying model in the local region specified in the premise the objective of structure identification is to identify an optimal partition of the input space into gaussian axis orthogonal fuzzy sets the linear parameters in the rule consequent are then estimated by means of a local weighted least squares algorithm lolimot is an incremental tree construction algorithm that partitions the input space by axis orthogonal splits in each iteration it greedily adds the new model that minimizes the classification error gp performs a global search for the optimal partition tree and is therefore able to backtrack in case of sub optimal intermediate split decisions we compare the performance of both methods for function approximation of a highly non linear two dimensional test function and an engine characteristic map

casimir operators of semidirect products of semisimple with heisenberg groups casimir operators for semidirect products of some semisimple groups with heisenberg groups are computed the analysis is carried out using dual representations on fock space wherein the action of the semidirect products are related to their dual groups namely certain unitary orthogonal and symplectic groups the compact symplectic group chain is also investigated by passing to the complexification groups between the symplectic groups are constructed which are of the form of semidirect products of symplectic groups with heisenberg groups

representations of sn u n in repeated tensor products of the unitary groups the n fold tensor product space generated by a given irreducible representation of the unitary group u n is a representation space for the symmetric group sn as well as for u n using ideas from the theory of dual pairs such tensor product spaces are decomposed into irreducible representations of u n times reducible representations of sn computationally effective formulae for the multiplicity of irreducible representations of sn are given generating sets of invariant polynomials from the enveloping algebra of u n that commute with the sn action are exhibited it is shown that the eigenvalues of such operators can be used to break the multiplicity occurring in the dual u n xsn action

on noncommutative multi solitons solitons in noncommutative scalar field theories possess many stringy features in particular multi solitons in these theories behave like multiple d branes a useful moduli space description for multi solitons exists which reproduces stringy features of d brane moduli spaces in a simple way via properties of special projection operators in particular they are equivalent to a hilbert scheme resolution of symmetric product spaces