character generators for elements of finite order in simple lie groups a1 a2 a3 b2 and g2 elements of finite order in the compact simple lie groups su 2 su 3 su 4 sp 4 or o 5 and g 2 are considered we provide the characters of the elements on irreducible representations of lie groups by assigning appro priate numerical values to the variables on which the characters of representations of the lie group depend in this way we specialize generating functions for the characters of the representations of the lie groups to the generating functions for characters of the elements of finite order particular attention is paid to rational elements all of whose characters are integers they are listed and the generating functions for their characters are obtained in a simplified form from which the characters can be read gaussian elements are also studied in detail their characters are complex valued with integer real and imaginary parts

on line fuzzy modeling via clustering and support vector machines in this paper we propose a novel approach to identify unknown nonlinear systems with fuzzy rules and support vector machines our approach consists of four steps which are on line clustering structure identification parameter identification and local model combination the collected data are firstly clustered into several groups through an on line clustering technique then structure identification is performed on each group using support vector machines such that the fuzzy rules are automatically generated with the support vectors time varying learning rates are applied to update the membership functions of the fuzzy rules the modeling errors are proven to be robustly stable with bounded uncertainties by a lyapunov method and an input to state stability technique comparisons with other related works are made through a real application of crude oil blending process the results demonstrate that our approach has good accuracy and this method is suitable for on line fuzzy modeling

radial basis function based adaptive fuzzy systems and their applications to system identification and prediction abstract in this paper we describe a neuro fuzzy system with adaptive capability to extract fuzzy if then rules from input and output sample data through learning the proposed system called radial basis function rbf based adaptive fuzzy system afs employs the gaussian functions to represent the membership functions of the premise part of fuzzy rules three architectural deviations of the rbf based afs are also presented according to different consequence types such as constant first order linear function and fuzzy variable these provide versatility of the network to handle arbitrary fuzzy inference schemes we present examples of system identification and time series prediction to illustrate how to solve these problems and to demonstrate its validity and effectiveness using the rbf based afs

on lie gradings ii abstract to obtain fine gradings of gl n c it is necessary to study maximal abelian subgroups mad subgroups of diagonizable automorphisms in a ut gl n c we described two finite sets h n and k n such that each mad subgroup consisting only from inner automorphisms is conjugated to some element of h n and each mad subgroup containing at least one outer automorphism is conjugated to some element of k n using these results concerning k n we then easily find all mad subgroups in a ut o n c for n 8 and in a ut sp 2n c

graded contractions of representations of orthogonal and symplectic lie algebras with respect to their maximal parabolic subalgebras parabolic gradings of the classical simple lie algebras o n c n or 5 and sp 2n c n or 2 with complex parameters are described for all maximal parabolic subalgebras all contractions which leave a maximal parabolic subalgebra intact and which preserve a parabolic grading parabolic contractions of lie algebras are found contractions of the irreducible representations for each parabolic contraction of the lie algebra are the main results of the article

scalable tsk fuzzy modeling for very large datasets using minimal enclosing ball approximation in order to overcome the difficulty in takagi sugeno kang tsk fuzzy modeling for large datasets scalable tsk stsk fuzzy model training is investigated in this study based on the core set based minimal enclosing ball meb approximation technique the specified l2 norm penalty based insensitive criterion is first proposed for tsk model training and it is found that such tsk fuzzy model training can be equivalently expressed as a center constrained meb problem with this finding an stsk fuzzy model training algorithm which is called stsk for large or very large datasets is then proposed by using the core set based meb approximation technique the proposed algorithm has two distinctive advantages over classical tsk fuzzy model training algorithms the maximum space complexity for training is not reliant on the size of the training dataset and the maximum time complexity for training is linear with the size of the training dataset as confirmed by extensive experiments on both synthetic and real world regression datasets

stationary fuzzy fokker planck learning and stochastic fuzzy filtering the application of nonlinear optimization to the estimation of fuzzy model parameters is well known to do the reverse of this the concept of stationary fuzzy fokker planck learning sffpl is introduced i e sffpl applies the fuzzy modeling technique in nonlinear optimization problems sffpl is based on the fuzzy approximation of the stationary cumulative distribution function of a stochastic search process associated with the nonlinear optimization problem a carefully designed algorithm is suggested for sffpl to locate the optimum point this paper also considers the variational bayes vb based inference of a stochastic fuzzy filter whose consequents as well as antecedents are random variables the problem of vb inference of stochastic antecedents because of the nonlinearity of the likelihood function is analytically intractable the sffpl algorithm for high dimensional nonlinear optimization that does not require the derivative of the objective function can be used to numerically solve the stochastic fuzzy filtering problem

multicomplementary operators via finite fourier transform a complete set of d 1 mutually unbiased bases exists in a hilbert space of dimension d whenever d is a power of a prime we discuss a simple construction of d 1 disjoint classes each one having d 1 commuting operators such that the corresponding eigenstates form sets of unbiased bases such a construction works properly for prime dimension we investigate an alternative construction in which the real numbers that label the classes are replaced by a finite field having d elements one of these classes is diagonal and can be mapped to cyclic operators by means of the finite fourier transform which allows one to understand complementarity in a similar way as for the position momentum pair in standard quantum mechanics the relevant examples of two and three qubits and two qutrits are discussed in detail

on maximum likelihood fuzzy neural networks in this paper m estimators where m stands for maximum likelihood used in robust regression theory for linear parametric regression problems will be generalized to nonparametric maximum likelihood fuzzy neural networks mfnns for nonlinear regression problems emphasis is put particularly on the robustness against outliers this provides alternative learning machines when faced with general nonlinear learning problems simple weight updating rules based on gradient descent and iteratively reweighted least squares irls will be derived some numerical examples will be provided to compare the robustness against outliers for usual fuzzy neural networks fnns and the proposed mfnns simulation results show that the mfnns proposed in this paper have good robustness against outliers

neuro fuzzy system with learning tolerant to imprecision in this paper a new learning method tolerant to imprecision is introduced and used in neuro fuzzy modeling this method can be called e insensitive learning where in order to fit the fuzzy model to real data a weighted e insensitive loss function is used the proposed method makes it possible to exclude an intrinsic inconsistency of neuro fuzzy modeling where zero tolerance learning is used to obtain a fuzzy model tolerant to imprecision the e insensitive learning leads to a model with the minimal vapnik chervonenkis dimension complexity which results in improving generalization ability of this system and its robustness to outliers finally numerical examples are given to demonstrate the validity of the introduced method

maps preserving peripheral spectrum of generalized products of operators abstract let a 1 and a 2 be standard operator algebras on complex banach spaces x 1 and x 2 respectively for k 2 let i 1 i m be a sequence with terms chosen from 1 k and assume that at least one of the terms in i 1 i m appears exactly once define the generalized product t 1 t 2 t k t i 1 t i 2 t i m on elements in a i let a 1 a 2 be a map with the range containing all operators of rank at most two we show that satisfies that a 1 a k a 1 a k for all a 1 a k where a stands for the peripheral spectrum of a if and only if is an isomorphism or an anti isomorphism multiplied by an mth root of unity and the latter case occurs only if the generalized product is quasi semi jordan if x 1 h and x 2 k are complex hilbert spaces we characterize also maps preserving the peripheral spectrum of the skew generalized products and prove that such maps are of the form a c u a u or a c u a t u where u b h k is a unitary operator c 1 1

support vector interval regression networks for interval regression analysis in this paper the support vector interval regression networks svirns are proposed for the interval regression analysis the svirns consist of two radial basis function networks one network identifies the upper side of data interval and the other network identifies the lower side of data intervals because the support vector regression svr approach is equivalent to solving a linear constrained quadratic programming problem the number of hidden nodes and the initial values of adjustable parameters can be easily obtained since the selection of a parameter e in the svr approach may seriously affect the modeling performance a two step approach is proposed to properly select the e value after the svr approach with the selected e an initial structure of svirns can be obtained besides outliers will not significantly affect the upper and lower bound interval obtained through the proposed two step approach consequently a traditional back propagation bp learning algorithm can be used to adjust the initial structure networks of svirns under training data sets without or with outliers due to the better initial structure of svirns are obtained by the svr approach the convergence rate of svirns is faster than the conventional networks with bp learning algorithms or with robust bp learning algorithms for interval regression analysis four examples are provided to show the validity and applicability of the proposed svirns

fine gradings on simple classical lie algebras abstract the fine abelian group gradings on the simple classical lie algebras including d 4 over algebraically closed fields of characteristic 0 are determined up to equivalence this is achieved by assigning certain invariant to such gradings that involve central graded division algebras and suitable sesquilinear forms on free modules over them

global anomalies on orbifolds we consider global anomalies for heterotic string theory formulated on orbifolds the vanishing of certain characteristic classes in group cohomology provides sufficient conditions for the absence of global anomalies for abelian orbifolds level matching implies these cohomology conditions so suffices for the absence of anomalies for nonabelian orbifolds level matching does not suffice and there are additional constraints we give some examples to illustrate these new constraints

bayesian takagi sugeno kang fuzzy classifier in this paper the takagi sugeno kang tsk fuzzy classifier is casted into the bayesian inference framework and a new fuzzy classifier called bayesian tsk fuzzy classifier b tsk fc is proposed accordingly the proposed classifier can be constructed by learning both the antecedent and consequent parameters of the involved fuzzy rules simultaneously as a result of the introduction of bayesian inference the proposed b tsk fc can be distinguished as follows 1 unlike most existing tsk fuzzy classifiers where the antecedent and consequent parameters of fuzzy rules are learnt in a decoupled manner and the antecedent parameters are learnt only in the input space the antecedent parameters of the fuzzy rules in b tsk fc are learnt by developing a fuzzy clustering method in the input output space and the consequent parameters of fuzzy rules are learnt in accordance with the maximum margin of separation principle thereby resulting to form an intrinsic link between the input and output spaces to achieve improved classification performance and better interpretability 2 with a dirichlet prior assumption about fuzzy memberships in fuzzy clustering a markov chain monte carlo technique is employed to estimate the parameters of the proposed classifier from a sampling perspective 3 rather than being rivals fuzziness and probability in b tsk fc are collaboratively modeled to enhance the performance of tsk fuzzy classifier in terms of classification and interpretability our experimental results in synthetic datasets as well as several real world datasets confirm such merits of the proposed fuzzy classifier

generalized graph theory on lie g and applications in conformal field theory each magic basis of lie g supports a generalized graph theory on lie g whose axioms include the graphs of order n as a special case on so n other examples include the sine area graphs of su n and the sine area graphs of su ini the set of all magic basis generalized graph theory pairs is not known but each possible pair corresponds to a large set of new conformal field theories

improving the interpretability of tsk fuzzy models by combining global learning and local learning the fuzzy inference system proposed by takagi sugeno and kang known as the tsk model in fuzzy system literature provides a powerful tool for modeling complex nonlinear systems unlike conventional modeling where a single model is used to describe the global behavior of a system tsk modeling is essentially a multimodel approach in which simple submodels typically linear models are combined to describe the global behavior of the system most existing learning algorithms for identifying the tsk model are based on minimizing the square of the residual between the overall outputs of the real system and the identified model although these algorithms can generate a tsk model with good global performance i e the model is capable of approximating the given system with arbitrary accuracy provided that sufficient rules are used and sufficient training data are available they cannot guarantee the resulting model to have a good local performance often the submodels in the tsk model may exhibit an erratic local behavior which is difficult to interpret since one of the important motivations of using the tsk model also other fuzzy models is to gain insights into the model it is important to investigate the interpretability issue of the tsk model we propose a new learning algorithm that integrates global learning and local learning in a single algorithmic framework this algorithm uses the idea of local weighed regression and local approximation in nonparametric statistics but remains the component of global fitting in the existing learning algorithms the algorithm is capable of adjusting its parameters based on the user s preference generating models with good tradeoff in terms of global fitting and local interpretation we illustrate the performance of the proposed algorithm using a motorcycle crash modeling example

hierarchical mixtures of experts and the em algorithm we present a tree structured architecture for supervised learning the statistical model underlying the architecture is a hierarchical mixture model in which both the mixture coefficients and the mixture components are generalized linear models glim s learning is treated as a maximum likelihood problem in particular we present an expectation maximization em algorithm for adjusting the parameters of the architecture we also develop an on line learning algorithm in which the parameters are updated incrementally comparative simulation results are presented in the robot dynamics domain

the eight fine gradings of sl 4 c and o 6 c a grading of a lie algebra is called fine if it cannot be further refined fine gradings provide basic information about the structure of the algebra there are eight fine gradings of the simple lie algebra of type a3 over the complex number field one of them root decomposition is the main tool of the theory and applications in working with a3 and with its representations one other has also been used in the literature and the rest have apparently not been recognized so far an explicit description of all the fine gradings of a3 is given in terms of the four dimensional sl 4 c and six dimensional orthogonal o 6 c representations of the algebra these results should be useful generally for choosing bases which reflect structural properties of the lie algebra for defining various sets of additive quantum numbers for systems with such symmetries and for systematic study of grading preserving contractions of this lie algebra

fuzzy prediction of chaotic time series based on singular value decomposition for dynamic systems with complex ill conditioned or nonlinear characteristics the modeling method based on fuzzy sets is very effective to describe the properties of the systems in this paper a fuzzy modeling method based on singular value decomposition svd is proposed first a fuzzy clustering method is used to confirm the input space of fuzzy model then the recursive least square algorithm with singular value decomposition is applied to estimate the consequent parameters of fuzzy model in order to avoid error delivery and error accumulation furthermore the parameters of fuzzy model are also optimized by the presented algorithm to demonstrate the performance of this modeling method simulations on mackey glass time series and lorenz chaotic system are performed the results show that this method provides effective and accurate prediction

the pauli matrices in n dimensions and finest gradings of simple lie algebras of type a n 1 properties of the lie algebra gl n c are described for a basis which is a generalization of the 2 2 pauli matrices the 3 3 case is described in detail the remarkable properties of that basis are the grading of the lie algebra it offers each grading subspace is one dimensional and the matrix group it generates it is a finite group with the center of sl n c as its commutator group

infinite dimensional algebras and a trigonometric basis for the classical lie algebras this paper explores features of the infinite dimensional algebras that have been previously introduced in particular it is shown that the classical simple lie algebras an bn cn dn may be expressed in an egalitarian basis with trigonometric structure constants the transformation to the standard cartan weyl basis and the particularly transparent n limit that this formulation allows is provided

fine grading of sl p2 c generated by tensor product of generalized pauli matrices and its symmetries study of the normalizer of the mad group corresponding to a fine grading offers the most important tool for describing symmetries in the system of nonlinear equations connected with contraction of a lie algebra one fine grading that is always present in any lie algebra sl n c is the pauli grading the mad group corresponding to it is generated by generalized pauli matrices for such mad group we already know its normalizer its quotient group is isomorphic to the lie group sl 2 zn z2 in this paper we deal with a more complicated situation namely that the fine grading of sl p2 c is given by a tensor product of the pauli matrices of the same order p p being a prime we describe the normalizer of the corresponding mad group and we show that its quotient group is isomorphic to sp 4 fp z2 where fp is the finite field with p elements

fuzzy support vector machines a support vector machine svm learns the decision surface from two distinct classes of the input points in many applications each input point may not be fully assigned to one of these two classes in this paper we apply a fuzzy membership to each input point and reformulate the svms such that different input points can make different contributions to the learning of decision surface we call the proposed method fuzzy svms fsvms

variations on a theme of heisenberg pauli and weyl the parentage between weyl pairs the generalized pauli group and the unitary group is investigated in detail we start from an abstract definition of the heisenberg weyl group on the field and then switch to the discrete heisenberg weyl group or generalized pauli group on a finite ring the main characteristics of the latter group an abstract group of order d3 noted pd are given conjugacy classes and irreducible representation classes or equivalently lie algebra of dimension d3 associated with pd leaving the abstract sector a set of weyl pairs in dimension d is derived from a polar decomposition of su 2 closely connected to angular momentum theory then a realization of the generalized pauli group pd and the construction of generalized pauli matrices in dimension d are revisited in terms of weyl pairs finally the lie algebra of the unitary group u d is obtained as a subalgebra of the lie algebra associated with pd this leads to a development of the lie algebra of u d in a basis consisting of d2 generalized pauli matrices in the case where d is a power of a prime integer the lie algebra of su d can be decomposed into d 1 cartan subalgebras

multi task tsk fuzzy system modeling using inter task correlation information the classical fuzzy system modeling methods have been typically developed for the single task modeling scene which is essentially not in accordance with many practical applications where a multi task problem must be considered for the given modeling task although a multi task problem can be decomposed into many single task sub problems the modeling results indeed tell us that the individual modeling approach will not be very suitable for multi task problems due to the ignorance of the inter task latent correlation between different tasks in order to circumvent this shortcoming a multi task takagi sugeno kang fuzzy system model is proposed based on the classical l2 norm takagi sugeno kang fuzzy system in this paper the proposed model cannot only take advantage of independent information of each task but also make use of the inter task latent correlation information effectively resulting to obtain better generalization performance for the built fuzzy systems experiments on synthetic and real world datasets demonstrate the applicability and distinctive performance of the proposed multi task fuzzy system model in multi task modeling scenarios

a kernel fuzzy c means clustering based fuzzy support vector machine algorithm for classification problems with outliers or noises the support vector machine svm has provided higher performance than traditional learning machines and has been widely applied in real world classification problems and nonlinear function estimation problems unfortunately the training process of the svm is sensitive to the outliers or noises in the training set in this paper a common misunderstanding of gaussian function based kernel fuzzy clustering is corrected and a kernel fuzzy c means clustering based fuzzy svm algorithm kfcm fsvm is developed to deal with the classification problems with outliers or noises in the kfcm fsvm algorithm we first use the fcm clustering to cluster each of two classes from the training set in the high dimensional feature space the farthest pair of clusters where one cluster comes from the positive class and the other from the negative class is then searched and forms one new training set with membership degrees finally we adopt fsvm to induce the final classification results on this new training set the computational complexity of the kfcm fsvm algorithm is analyzed a set of experiments is conducted on six benchmarking datasets and four artificial datasets for testing the generalization performance of the kfcm fsvm algorithm the results indicate that the kfcm fsvm algorithm is robust for classification problems with outliers or noises

an angular momentum approach to quadratic fourier transform hadamard matrices gauss sums mutually unbiased bases the unitary group and the pauli group the construction of unitary operator bases in a finite dimensional hilbert space is reviewed through a nonstandard approach combining angular momentum theory and representation theory of su 2 a single formula for the bases is obtained from a polar decomposition of su 2 and is analyzed in terms of cyclic groups quadratic fourier transforms hadamard matrices and generalized gauss sums weyl pairs generalized pauli operators and their application to the unitary group and the pauli group naturally arise in this approach

sl 3 c generator matrix elements in a pauli subgroup basis we construct sl 3 c basis states reduced according to its finite subgroup 3 matrix elements of sl 3 c generators are calculated between 3 basis states

twisted cocycles of lie algebras and corresponding invariant functions abstract we consider finite dimensional complex lie algebras using certain complex parameters we generalize the concept of cohomology cocycles of lie algebras a special case is generalization of 1 cocycles with respect to the adjoint representation so called derivations parametric sets of spaces of cocycles allow us to define complex functions which are invariant under lie isomorphisms such complex functions thus represent useful invariants we show how they classify three and four dimensional lie algebras as well as how they apply to some eight dimensional one parametric nilpotent continua of lie algebras these functions also provide necessary criteria for existence of 1 parametric continuous contraction

structure identification of fuzzy model abstract the problems of structure identification of a fuzzy model are formulated a criterion for the verification of a structure is discussed using the criterion an algorithm for identifying a structure is suggested further a successive identification algorithm of the parameters is suggested the proposed methods are applied to an example

a clustering algorithm for fuzzy model identification the fuzzy model proposed by takagi and sugeno can represent highly nonlinear systems and is widely used for the representation of fuzzy rules in this paper the model is firstly modified to make its identification easier based on the fuzzy c partition space four criteria are proposed for optimization of the model parameters following that a clustering algorithm composed of fuzzy c linear functions clustering and like fuzzy c means clustering is developed for minimizing the four criteria an identification scheme for rule s premise and consequence parameters is deduced from the clustering algorithm in succession finally four examples are demonstrated to verify the effectiveness of the proposed algorithm

on the monomiality of nice error bases unitary error bases generalize the pauli matrices to higher dimensional systems two basic constructions of unitary error bases are known an algebraic construction by knill that yields nice error bases and a combinatorial construction by werner that yields shift and multiply bases an open problem posed by schlingemann and werner relates these two constructions and asks whether each nice error basis is equivalent to a shift and multiply basis we solve this problem and show that the answer is negative however we find that nice error bases have more structure than one can anticipate from their definition in particular we show that nice error bases can be written in a form in which at least half of the matrix entries are 0

fuzzy c p means clustering and its application to a fuzzy rule based classifier toward good generalization and good interpretability this paper introduces a new classifier design method based on a modification of the classical fuzzy c means clustering first a new fuzzy c means clustering with p constant prototypes is proposed this method can be considered a generalization of the concept of the conditional fuzzy clustering with some prototypes a priori known a special initialization of the prototypes is introduced next the proposed clustering method is used to construct the premises of an if then rule based classifier the conclusions of these rules are obtained by minimization of a criterion function with various approximations of a misclassification error e g based on the quadratic the linear the sigmoidal or the huber s loss function the conjugate gradient algorithm is used to minimize the proposed criterion function each if then rule is represented in the mamdani assilan form which has good interpretability finally an extensive experimental analysis on 14 benchmark datasets is performed to demonstrate the validity of the classifier introduced its competitiveness to the state of the art classifiers with respect to both performance and interpretability is also shown

generalized graphs and unitary irrational central charge in the superconformal master equation for each magic basis of lie g it is known that the virasoro master equation on affine g contains a generalized graph theory of conformal level families in this paper it is found that the superconformal master equation on affine g so dim g similarly contains a generalized graph theory of superconformal level families for each magic basis of g the superconformal level families satisfy linear equations on the generalized graphs and the first exact unitary irrational solutions of the superconformal master equation are obtained on the sine area graphs of g su n including the simplest unitary irrational central charges c 6nx nx 8 sin2 rs n yet observed in the program

new gradings of sl 3 c representations two new gradings of the lie algebra sl 3 c are the refined sl 2 c and o 3 gradings the grading in each case utilizes the sl 2 c or o 3 weight together with a new additive modular label a complete set of sl 3 c representation basis states labeled by each to the two sets of additive quantum numbers is found the newlabeling operator in both cases fails to commute with the cubic sl 3 c casimir operator and hence mixes states of the contragredient representations p q and q p

a transformed input domain approach to fuzzy modeling this paper presents an explanation of a fuzzy model considering the correlation among components of input data generally fuzzy models have a capability of dividing an input space into several subspaces compared to a linear model but hitherto suggested fuzzy modeling algorithms have not taken into consideration the correlation among components of sample data and have addressed them independently which results in an ineffective partition of the input space in order to solve this problem this paper proposes a new fuzzy modeling algorithm which partitions the input space more effectively than conventional fuzzy modeling algorithms by taking into consideration the correlation among components of sample data as a way to use the correlation and divide the input space the method of principal component is used finally the results of the computer simulation are given to demonstrate the validity of this algorithm

support vector learning for fuzzy rule based classification systems to design a fuzzy rule based classification system fuzzy classifier with good generalization ability in a high dimensional feature space has been an active research topic for a long time as a powerful machine learning approach for pattern recognition problems the support vector machine svm is known to have good generalization ability more importantly an svm can work very well on a high or even infinite dimensional feature space this paper investigates the connection between fuzzy classifiers and kernel machines establishes a link between fuzzy rules and kernels and proposes a learning algorithm for fuzzy classifiers we first show that a fuzzy classifier implicitly defines a translation invariant kernel under the assumption that all membership functions associated with the same input variable are generated from location transformation of a reference function fuzzy inference on the if part of a fuzzy rule can be viewed as evaluating the kernel function the kernel function is then proven to be a mercer kernel if the reference functions meet a certain spectral requirement the corresponding fuzzy classifier is named positive definite fuzzy classifier pdfc a pdfc can be built from the given training samples based on a support vector learning approach with the if part fuzzy rules given by the support vectors since the learning process minimizes an upper bound on the expected risk expected prediction error instead of the empirical risk training error the resulting pdfc usually has good generalization moreover because of the sparsity properties of the svms the number of fuzzy rules is irrelevant to the dimension of input space in this sense we avoid the curse of dimensionality finally pdfcs with different reference functions are constructed using the support vector learning approach the performance of the pdfcs is illustrated by extensive experimental results comparisons with other methods are also provided

lie product and local spectrum preservers abstract let x and y be two infinite dimensional complex banach spaces and fix two nonzero vectors x 0 x and y 0 y let b x resp b y denote the algebra of all bounded linear operators on x resp on y and let e x 0 x be the collection of all operators t b x for which x 0 is an eigenvector and t r 1 x 2 is a nonzero scalar operator for some scalar r c we show that a map from b x onto b y satisfies t s s t y 0 t s s t x 0 t s b x if and only if there are two functions b x c and b x 1 1 and a bijective bounded linear mapping a x y such that a x 0 y 0 the function is constant on b x e x 0 x and t t a t a 1 t 1 y for all t b x

a hopf algebra based on the patera zassenhaus grading of an the lie algebra gl n c admits a basis which is a generalization of the pauli spin operators as described by patera and zassenhaus 1988 in this paper a new expression for the set of generators of the enveloping algebra of gl n c is deduced and new quasi triangular hopf algebras with solutions to the quantum yang baxter equation for gl n c are derived solutions in eight vertex form are also given properties of the algebras suggest applications to physical theories

a locally recurrent fuzzy neural network with support vector regression for dynamic system modeling this paper proposes a new recurrent model known as the locally recurrent fuzzy neural network with support vector regression lrfnn svr that handles problems with temporal properties structurally an lrfnn svr is a five layered recurrent network the recurrent structure in an lrfnn svr comes from locally feeding the firing strength of each fuzzy rule back to itself the consequent layer in an lrfnn svr is a takagi sugeno kang t s k type consequent which is a linear function of current states regardless of system input and output delays for the structure learning a one pass clustering algorithm clusters the input training data and determines the number of network nodes in hidden layers for the parameter learning an iterative linear svr algorithm is proposed to tune free parameters in the rule consequent part and feedback loops the motivation for using svr for parameter learning is to improve the lrfnn svr generalization ability this paper demonstrates lrfnn svr capabilities by conducting simulations in dynamic system prediction and identification problems with noiseless and noisy data in addition this paper compares simulation results from the lrfnn svr with other recurrent fuzzy models

on lie gradings iii gradings of the real forms of classical lie algebras abstract maximal abelian subgroups of diagonalizable automorphisms of lie algebra so called mad groups play a crucial role in the construction of fine gradings of lie algebra our aim is to give a description of mad groups for real forms of classical lie algebras we introduce four types of matrix subgroups of g l n c called out groups ad groups out groups and ad groups for each type of these subgroups we define a relation of equivalence the problem of classifying of all non conjugate mad groups on real forms of sl n c o n c or sp n c is transformed to the problem of classifying these equivalence classes the classification of these equivalence classes is presented here

gradings on simple jordan and lie algebras abstract in this paper we describe all group gradings by a finite abelian group g of several types of simple jordan and lie algebras over an algebraically closed field f of characteristic zero

weighted least squares training of support vector classifiers leading to compact and adaptive schemes an iterative block training method for support vector classifiers svcs based on weighted least squares wls optimization is presented the algorithm which minimizes structural risk in the primal space is applicable to both linear and nonlinear machines in some nonlinear cases it is necessary to previously find a projection of data onto an intermediate dimensional space by means of either principal component analysis or clustering techniques the proposed approach yields very compact machines the complexity reduction with respect to the svc solution is especially notable in problems with highly overlapped classes furthermore the formulation in terms of wls minimization makes the development of adaptive svcs straightforward opening up new fields of application for this type of model mainly online processing of large amounts of static stationary data as well as online update in nonstationary scenarios adaptive solutions the performance of this new type of algorithm is analyzed by means of several simulations

spectrum preserving linear maps abstract let x and y be banach spaces we show that a spectrum preserving surjective linear map from b x to b y is either of the form t ata 1 for an isomorphism a of x onto y or the form t bt b 1 for an isomorphism b of x onto y

finite mode analogs of 2d ideal hydrodynamics coadjoint orbits and local canonical structure abstract the algebraic finite mode hydrodynamic type systems that have o n integrals of motion for o n n modes and are intrinsically connected with two dimensional ideal fluid flows are studied special emphasis is given on the structure of their phase space and the problem of finding proper dynamical variables for its description in connection with appropriate problems for the original hydrodynamic equations various possibilities for embedding in the space of canonical variables are demonstrated on the simplest example of the su 3 model giving the finite dimensional analogs of clebsch variables

maps preserving the spectrum of polynomial products of matrices abstract let m p c be the algebra of all p p complex matrices and let t denote the spectrum of any matrix t m p c for an integer n 2 let f 1 n be a 1 linear polynomial with complex coefficients and n non commuting indeterminates 1 n under minor natural conditions on f we show that if a map on m p c satisfies 0 1 f t 1 t n f t 1 t n for all t 1 t n m p c then there exist a nonzero scalar and an invertible matrix a m p c such that has one of the following forms 0 2 t a t a 1 for all t m p c or 0 3 t a t t a 1 for all t m p c here t t denotes the transpose of t in general not all of mappings of the form 0 2 or the form 0 3 satisfy 0 1 when n 2 we describe the set of all scalars for which the mapping 0 2 or 0 3 satisfies 0 1 we also obtain analogue results of spectrum preserving maps on the real linear space h p of all self adjoint matrices in m p c our results extend and unify several results obtained earlier on maps on m p c preserving the spectrum of generalized product or generalized jordan product of matrices

fine gradings of o 5 c sp 4 c and of their real forms there are three fine gradings of the simple lie algebra of type b2 over the complex number field they provide a basic information about the structure of the algebra in the paper an explicit description of all fine gradings is given in terms of the four dimensional symplectic sp 4 c and five dimensional orthogonal o 5 c representations of the algebra in addition the real forms of b2 are considered it is shown which of the fine gradings survive the restriction to each of the real forms these results should be useful in defining various sets of additive quantum numbers for systems with such symmetries for systematic study of grading preserving contractions of this lie algebra and generally for choosing bases which reflect structural properties of the lie algebra

variational bayes for a mixed stochastic deterministic fuzzy filter this study under the variational bayes vb framework infers the parameters of a takagi sugeno fuzzy filter having deterministic antecedents and stochastic consequents the aim of this study is to take advantages of the vb framework to design fuzzy filtering algorithms which include an automated regularization incorporation of statistical noise models and model comparison capability the vb method can be easily applied to the linear in parameters models this paper applies the vb method to the nonlinear fuzzy filters without using taylor expansion for a linear approximation of some nonlinear function it is assumed that the nonlinear parameters i e antecedents of the fuzzy filter are deterministic while linear parameters are stochastic the vb algorithm by maximizing a strict lower bound on the data evidence makes the approximate posterior of linear parameters as close to the true posterior as possible the nonlinear deterministic parameters are tuned in a way to further increase the lower bound on data evidence the vb paradigm can be used to design an algorithm that automatically selects the most suitable fuzzy filter out of the considered finite set of fuzzy filters this is done by fitting the observed data as a stochastic combination of the different takagi sugeno fuzzy filters such that the individual filters compete with one another to model the data

towards a robust fuzzy clustering fuzzy clustering helps to find natural vague boundaries in data the fuzzy c means method fcm is one of the most popular clustering methods based on minimization of a criterion function however one of the greatest disadvantages of this method is its sensitivity to presence of noise and outliers in data this paper introduces a new e insensitive fuzzy c means efcm clustering algorithm as a special case this algorithm includes the well known fuzzy c medians method fcmed also methods with insensitivity control named fcm and fcm are introduced performance of the new clustering algorithm is experimentally compared with the fcm method using synthetic data with outliers and heavy tailed and overlapped groups of data in background noise

on the structure of graded lie algebras we study the structure of arbitrary graded lie algebras we show that any of such algebras l with a symmetric g support is of the form l u jij with u a subspace of l1 and any ij a well described graded ideal of l satisfying ij ik 0 if j k under certain conditions the gr simplicity of l is characterized and it is shown that l is the direct sum of the family of its minimal graded ideals each one being a gr simple graded lie algebra

a new kernel based fuzzy clustering approach support vector clustering with cell growing in this paper the support vector clustering is extended to an adaptive cell growing model which maps data points to a high dimensional feature space through a desired kernel function this generalized model is called multiple spheres support vector clustering which essentially identifies dense regions in the original space by finding their corresponding spheres with minimal radius in the feature space a multisphere clustering algorithm based on adaptive cluster cell growing method is developed whereby it is possible to obtain the grade of memberships as well as cluster prototypes in partition the effectiveness of the proposed algorithm is demonstrated for the problem of arbitrary cluster shapes and for prototype identification in an actual application to a handwritten digit data set

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

graded contractions of the pauli graded sl 3 c the lie algebra sl 3 c is considered in the basis of generalized pauli matrices corresponding grading is the pauli grading here it is one of the four gradings of the algebra which cannot be further refined the set s of 48 contraction equations for 24 contraction parameters is solved our main tools are the symmetry group of the pauli grading of sl 3 c which is essentially the finite group sl 2 z3 and the induced symmetry of the system s a list of all equivalence classes of solutions of the contraction equations is provided among the solutions 175 equivalence classes are non parametric and 13 solutions depend on one or two continuous parameters providing a continuum of equivalence classes and subsequently continuum of non isomorphic lie algebras solutions of the contraction equations of pauli graded sl 3 c are identified here as specific solvable lie algebras of dimensions up to 8 earlier algorithms for identification of lie algebras given by their structure constants had to be made more efficient in order to distinguish non isomorphic lie algebras encountered here resulting lie algebras are summarized in tabular form there are 88 indecomposable solvable lie algebras of dimension 8 77 of them being nilpotent there are 11 infinite sets of parametric lie algebra which still deserve further study

fine gradings of o 4 c a grading of a lie algebra is called fine if it cannot be further refined fine gradings provide basic information about the structure of the algebra there are six fine gradings of the semisimple lie algebra of type a1 a1 over the complex number field an explicit description of all the fine gradings of a1 a1 is given in terms of the four dimensional representation o 4 c of the algebra

a new approach to fuzzy modeling this paper proposes a new approach to fuzzy modeling the suggested fuzzy model can express a given unknown system with a few fuzzy rules as well as takagi and sugeno s model 1985 because it has the same structure as that of takagi and sugeno s model it is also as easy to implement as sugeno and yasukawa s model 1993 because its identification mimics the simple identification procedure of sugeno and yasukawa s model the suggested algorithm is composed of two steps coarse tuning and fine tuning in coarse tuning fuzzy c regression model fcrm clustering is used which is a modified version of fuzzy c means fcm in fine tuning gradient descent algorithm is used to precisely adjust parameters of the fuzzy model instead of nonlinear optimization methods used in other models finally some examples are given to demonstrate the validity of this algorithm

a fuzzy system constructed by rule generation and iterative linear svr for antecedent and consequent parameter optimization this paper proposes a new fuzzy regression model i e the fuzzy system constructed by rule generation and iterative linear support vector regression fs rglsvr for structural risk minimization the fs rglsvr is composed of takagi sugeno ts type fuzzy if then rules these rules are automatically constructed by a self splitting rule generation algorithm that introduces the self splitting technique to the k means clustering algorithm this new algorithm regards a cluster as a fuzzy rule where no preassignment of the cluster rule number is necessary the cost function for parameter learning is defined based on structural risk instead of empirical risk minimization in order to achieve generalizability tuning all of the free parameters in the fs rglsvr using linear support vector regression svr is proposed to minimize the cost function each of the consequent and antecedent part parameters is expressed as a linear combination coefficient in a transformed input space so that the linear svr is applicable this paper introduces iterative linear svr to tune antecedent and consequent parameters this paper demonstrates the capabilities of fs rglsvr by two simulated and four practical regression examples comparisons with fuzzy systems with different types of learning algorithms verify the performance of the fs rglsvr

maps preserving the spectrum of generalized jordan product of operators abstract let a 1 a 2 be standard operator algebras on complex banach spaces x 1 x 2 respectively for k 2 let i 1 i m be a sequence with terms chosen from 1 k and define the generalized jordan product t 1 t k t i 1 t i m t i m t i 1 on elements in a i this includes the usual jordan product a 1 a 2 a 1 a 2 a 2 a 1 and the triple a 1 a 2 a 3 a 1 a 2 a 3 a 3 a 2 a 1 assume that at least one of the terms in i 1 i m appears exactly once let a map a 1 a 2 satisfy that a 1 a k a 1 a k whenever any one of a 1 a k has rank at most one it is shown in this paper that if the range of contains all operators of rank at most three then must be a jordan isomorphism multiplied by an m th root of unity similar results for maps between self adjoint operators acting on hilbert spaces are also obtained

neuro fuzzy rule generation survey in soft computing framework the present article is a novel attempt in providing an exhaustive survey of neuro fuzzy rule generation algorithms rule generation from artificial neural networks is gaining in popularity in recent times due to its capability of providing some insight to the user about the symbolic knowledge embedded within the network fuzzy sets are an aid in providing this information in a more human comprehensible or natural form and can handle uncertainties at various levels the neuro fuzzy approach symbiotically combining the merits of connectionist and fuzzy approaches constitutes a key component of soft computing at this stage to date there has been no detailed and integrated categorization of the various neuro fuzzy models used for rule generation we propose to bring these together under a unified soft computing framework moreover we include both rule extraction and rule refinement in the broader perspective of rule generation rules learned and generated for fuzzy reasoning and fuzzy control are also considered from this wider viewpoint models are grouped on the basis of their level of neuro fuzzy synthesis use of other soft computing tools like genetic algorithms and rough sets are emphasized rule generation from fuzzy knowledge based networks which initially encode some crude domain knowledge are found to result in more refined rules finally real life application to medical diagnosis is provided

t2fela type 2 fuzzy extreme learning algorithm for fast training of interval type 2 tsk fuzzy logic system a challenge in modeling type 2 fuzzy logic systems is the development of efficient learning algorithms to cope with the ever increasing size of real world data sets in this paper the extreme learning strategy is introduced to develop a fast training algorithm for interval type 2 takagi sugeno kang fuzzy logic systems the proposed algorithm called type 2 fuzzy extreme learning algorithm t2fela has two distinctive characteristics first the parameters of the antecedents are randomly generated and parameters of the consequents are obtained by a fast learning method according to the extreme learning mechanism in addition because the obtained parameters are optimal in the sense of minimizing the norm the resulting fuzzy systems exhibit better generalization performance the experimental results clearly demonstrate that the training speed of the proposed t2fela algorithm is superior to that of the existing state of the art algorithms the proposed algorithm also shows competitive performance in generalization abilities

pauli gradings on lie superalgebras and graded codimension growth abstract we introduce grading on certain finite dimensional simple lie superalgebras of type p t by elementary abelian 2 group this grading gives rise to pauli matrices and is a far generalization of z 2 z 2 grading on lie algebra of 2 2 traceless matrices we use this grading for studying numerical invariants of polynomial identities of lie superalgebras in particular we compute graded pi exponent corresponding to pauli grading

finest grading of the lie superalgebra gl n n the author shows that the group p2n of generalised pauli matrices in 2n dimensions provides a finest grading of the lie superalgebra gl n n and of its subalgebras sl n n and a n 1 n 1

group theoretical construction of mutually unbiased bases in hilbert spaces of prime dimensions mutually unbiased bases in hilbert spaces of finite dimensions are closely related to the quantal notion of complementarity an alternative proof of existence of a maximal collection of n 1 mutually unbiased bases in hilbert spaces of prime dimension n is given by exploiting the finite heisenberg group also called the pauli group and the action of on finite phase space implemented by unitary operators in the hilbert space crucial for the proof is that for prime is also a finite field

maps preserving zeros of a polynomial abstract let a be an algebra and let f x 1 x d be a multilinear polynomial in noncommuting indeterminates x i we consider the problem of describing linear maps a a that preserve zeros of f under certain technical restrictions we solve the problem for general polynomials f in the case where a m n f we also consider quite general algebras a but only for specific polynomials f

data driven elastic fuzzy logic system modeling constructing a concise system with human like inference mechanism the construction of fuzzy logic systems flss using data driven techniques has become the most popular modeling approach however this approach still faces critical challenges including the difficulty in obtaining concise models for high dimensional data and generating accurate fuzzy rules to simulate human inference mechanism to tackle these issues a new fls modeling framework called data driven elastic fls dd efls is proposed in this paper the dd efls has two key characteristics first the fuzzy rules in the rule base can use different feature subspaces that are extracted from the original high dimensional space to yield simple and accurate rules in feature spaces of lower dimensionality second fuzzy inferences from various views are implemented by embedding different rules in the corresponding subspaces to imitate human inference mechanism based on the dd efls framework an elastic takagi sugeno kang tsk fls modeling method etsk fls is proposed to train the elastic tsk fls using the concise rules and a more human like inference mechanism for modeling tasks based on high dimensional datasets the characteristics and advantages of the proposed framework and the etsk fls method are validated experimentally using both synthetic and real world datasets

fine group gradings of the real forms of sl 4 c sp 4 c and o 4 c we present an explicit description of the fine group gradings i e group gradings which cannot be further refined of the real forms of the semisimple lie algebras sl 4 c sp 4 c and o 4 c all together 12 real lie algebras are considered and a total of 44 of their fine group gradings are listed the inclusions sl 4 c sp 4 c o 4 c are an important tool in our presentation systematic use is made of the faithful representations of the three lie algebras by 4 4 matrices

tsk fuzzy modeling based on spl epsiv insensitive learning in this paper new learning methods tolerant to imprecision are introduced and applied to fuzzy modeling based on the takagi sugeno kang fuzzy system the fuzzy modeling has an intrinsic inconsistency it may perform thinking tolerant to imprecision but learning methods are zero tolerant to imprecision the proposed methods make it possible to exclude this intrinsic inconsistency of a fuzzy modeling where zero tolerance learning is used to obtain fuzzy model tolerant to imprecision these new methods can be called spl epsiv insensitive learning or spl epsiv learning where in order to fit the fuzzy model to real data the spl epsiv insensitive loss function is used this leads to a weighted or fuzzified version of vapnik s support vector regression machine this paper introduces two approaches to solving the spl epsiv insensitive learning problem the first approach leads to the quadratic programming problem with bound constraints and one linear equality constraint the second approach leads to a problem of solving a system of linear inequalities two computationally efficient numerical methods for the spl epsiv insensitive learning are proposed the spl epsiv insensitive learning leads to a model with the minimal vapnik chervonenkis dimension which results in an improved generalization ability of this model and its outliers robustness finally numerical examples are given to demonstrate the validity of the introduced methods

support vector fuzzy regression machines support vector machine svm has been very successful in pattern recognition and function estimation problems in this paper we introduce the use of svm for multivariate fuzzy linear and nonlinear regression models using the basic idea underlying svm for multivariate fuzzy regressions gives computational efficiency of getting solutions

the four sets of additive quantum numbers of su 3 the four maximal sets of additive quantum numbers and related fine gradings of the lie algebra sl 3 c are described in detail the quantum numbers are determined by a grading of the lie algebra fine gradings providing maximal sets of them two sets of additive quantum numbers are equivalent precisely if the corresponding gradings are equivalent under a transformation from the automorphism group of the lie algebra