determination of point group harmonics for arbitrary j by a projection method i cubic group quantization along an axis of order 4 we give a method for systematically building cubic harmonics i e base vectors of irreducible representations of the group of invariance of a cube applications include solid state physics molecular chemistry spectroscopy etc the only necessary operators are cz4 and ry rotation of 2 around 0y projectors on representations are constructed then their rows are orthogonalized by a schmidt method and normalized simply by dividing by the square root of a diagonal element

majorization as a tool for optimizing a class of matrix functions the problem of minimizing a general matrix trace function possibly subject to certain constraints is approached by means of majorizing this function by one having a simple quadratic shape and whose minimum is easily found it is shown that the parameter set that minimizes the majorizing function also decreases the matrix trace function which in turn provides a monotonically convergent algorithm for minimizing the matrix trace function iteratively three algorithms based on majorization for solving certain least squares problems are shown to be special cases in addition by means of several examples it is noted how algorithms may be provided for a wide class of statistical optimization tasks for which no satisfactory algorithms seem available

a foundation on semigroups of operators defined on the set of triangular fuzzy numbers and its application to fuzzy fractional evolution equations abstract in this article we study the strongly continuous semigroups of fuzzy valued operators defined on the space of triangular fuzzy numbers new notions of fuzzy infinitesimal generators and fuzzy resolvent operators of fuzzy semigroups are established in the sense of the generalized hukuhara difference a hille yosida like theorem in the framework of fuzzy metric space is investigated we demonstrate the efficiency of theoretical results by studying the existence of mild solutions of fuzzy nonlinear fractional evolution equations

minimization of a class of matrix trace functions by means of refined majorization a procedure is described for minimizing a class of matrix trace functions the procedure is a refinement of an earlier procedure for minimizing the class of matrix trace functions using majorization it contains a recently proposed algorithm by koschat and swayne for weighted procrustes rotation as a special case a number of trial analyses demonstrate that the refined majorization procedure is more efficient than the earlier majorization based procedure

simplimax oblique rotation to an optimal target with simple structure factor analysis and principal component analysis are usually followed by simple structure rotations of the loadings these rotations optimize a certain criterion e g varimax oblimin designed to measure the degree of simple structure of the pattern matrix simple structure can be considered optimal if a usually large number of pattern elements is exactly zero in the present paper a class of oblique rotation procedures is proposed to rotate a pattern matrix such that it optimally resembles a matrix which has an exact simple pattern it is demonstrated that this method can recover relatively complex simple structures where other well known simple structure rotation techniques fail

a survey on fuzzy fractional variational problems abstract this paper focuses on introducing and studying generalized fuzzy euler lagrange equation and fuzzy isoperimetric problem according to the concept of caputo type fuzzy fractional derivative in the sense of the generalized fuzzy differentiability we extend and establish some definitions on fuzzy fractional calculus of variation and provide some necessary conditions to obtain the fuzzy fractional euler lagrange equation for both constrained and unconstrained fuzzy fractional variational problems the fuzzy isoperimetric problem is also investigated in relation to the generalized fuzzy euler lagrange finally two examples are given to describe the proposed approach

existence results for extremal solutions of interval fractional functional integro differential equations abstract in this study we introduce interval fractional functional integro differential equations iffides under the caputo generalized hukuhara fractional differentiability we establish some necessary comparison results and use the monotone iterative technique combined with the method of upper and lower solutions to investigate the existence of extremal solutions for iffides an example is presented to illustrate the results

clustering of multivariate binary data with dimension reduction via l1 regularized likelihood maximization clustering methods with dimension reduction have been receiving considerable wide interest in statistics lately and a lot of methods to simultaneously perform clustering and dimension reduction have been proposed this work presents a novel procedure for simultaneously determining the optimal cluster structure for multivariate binary data and the subspace to represent that cluster structure the method is based on a finite mixture model of multivariate bernoulli distributions and each component is assumed to have a low dimensional representation of the cluster structure this method can be considered as an extension of the traditional latent class analysis sparsity is introduced to the loading values which produces the low dimensional subspace for enhanced interpretability and more stable extraction of the subspace an em based algorithm is developed to efficiently solve the proposed optimization problem we demonstrate the effectiveness of the proposed method by applying it to a simulation study and real datasets highlightswe propose a novel clustering method of multivariate binary data the proposed method provides a low dimensional representation of clusters the proposed method overcomes the conventional tandem analysis the proposed method conducts feature selection for clustering

determination of point group harmonics for arbitrary j by a projection method iii cubic group quantization along a ternary axis the method described in a first paper to obtain cubic harmonics quantized on an axis of order 4 is applied to the case of a ternary quantization axis projectors on irreducible representations are expressed with rotation matrices r 0 and r 0 arccos 1 3 acting on subspaces of su 2 invariant under d3 relations between the descriptions on the two axes of quantization are derived

fuzzy partial differential equations under the cross product of fuzzy numbers abstract this paper presents a new approach for obtaining triangular solutions of linear partial differential equations with fuzzy coefficients our approach is based on employing the concept of cross product of fuzzy numbers the benefit of our method is to simplify the complicated procedure arising from the term of the usual product of a fuzzy number by an unknown fuzzy function in equations we examine some properties related to the cross product of fuzzy numbers finally some examples are given to illustrate the theoretical results

sequential estimation of velocity fields using episodic proper orthogonal decomposition abstract in this paper the problem of approximating velocity fields at future and past times based on information available at the current time is addressed a novel method called episodic pod is described and developed that enables us to achieve our objective application of episodic pod to an ensemble of flow data results in a set of spatio temporal eigenfunctions and a set of coefficients associated with the eigenfunctions from these eigenfunctions we develop two models called the forward model and inverse model that enable us to approximate the velocity fields at future and past times based on information provided at the current time a second set of models the forward and inverse sequential models are also developed that enable the dynamic update of approximated velocity fields when new information is made available making these models more adept at on line estimation the various properties associated with these models are described in detail and four different examples are used to validate the models and show the different properties associated with episodic pod it is shown through numerical validation that the episodic pod model has a form that is dynamically consistent with the original system it is also shown that episodic pod outperforms linear kalman filters in the presence of noise

oblique rotaton in canonical correlation analysis reformulated as maximizing the generalized coefficient of determination to facilitate the interpretation of canonical correlation analysis cca solutions procedures have been proposed in which cca solutions are orthogonally rotated to a simple structure in this paper we consider oblique rotation for cca to provide solutions that are much easier to interpret though only orthogonal rotation is allowed in the existing formulations of cca our task is thus to reformulate cca so that its solutions have the freedom of oblique rotation such a task can be achieved using yanai s jpn j behaviormetrics 1 46 54 1974 j jpn stat soc 11 43 53 1981 generalized coefficient of determination for the objective function to be maximized in cca the resulting solutions are proved to include the existing orthogonal ones as special cases and to be rotated obliquely without affecting the objective function value where ten berge s psychometrika 48 519 523 1983 theorems on suborthonormal matrices are used a real data example demonstrates that the proposed oblique rotation can provide simple easily interpreted cca solutions

fuzzy fractional functional integral and differential equations this paper is devoted to considering the existence and uniqueness results for fuzzy fractional functional integral equations employing the contraction principle moreover the approach is followed to prove the existence and uniqueness of solutions for the fuzzy fractional functional initial value problem under caputo type fuzzy fractional derivatives finally the solution to fuzzy fractional functional initial value problem under caputo type fuzzy fractional derivatives by a modified fractional euler method mfem is presented the method is illustrated by solving some examples

generalized differentiability of fuzzy valued functions in the present paper using novel generalizations of the hukuhara difference for fuzzy sets we introduce and study new generalized differentiability concepts for fuzzy valued functions several properties of the new concepts are investigated and they are compared to similar fuzzy differentiabilities finding connections between them characterization and relatively simple expressions are provided for the new derivatives

mechanics with fractional derivatives lagrangian and hamiltonian mechanics can be formulated to include derivatives of fractional order f riewe phys rev 53 1890 1996 lagrangians with fractional derivatives lead directly to equations of motion with nonconservative classical forces such as friction the present work continues the development of fractional derivative mechanics by deriving a modified hamilton s principle introducing two types of canonical transformations and deriving the hamilton jacobi equation using generalized mechanics with fractional and higher order derivatives the method is illustrated with a frictional force proportional to velocity in contrast to conventional mechanics with integer order derivatives quantization of a fractional derivative hamiltonian cannot generally be achieved by the traditional replacement of momenta with coordinate derivatives instead a quantum mechanical wave equation is proposed that follows from the hamilton jacobi equation by application of the correspondence principle

on the solution of fuzzy fractional optimal control problems with the caputo derivative abstract this paper presents an extension to fractional optimal control problems with ambiguity as the ambiguity is modeled with fuzzy method we encounter with a fuzzy fractional optimal control problem the objective in fuzzy fractional optimal control problem is to determine the best possible fuzzy control which satisfies the related fuzzy fractional dynamic systems and minimizes the fuzzy performance index here the fractional derivative is described in the caputo sense to find the solution first we state some definitions and prove some required theorems then we employ the obtained result to determine the necessary conditions furthermore we show that the obtained necessary optimality conditions become sufficient by considering some extra assumptions finally some examples are presented for more illustration of the subject

quartic rotation criteria and algorithms most of the currently used analytic rotation criteria for simple structure in factor analysis are summarized and identified as members of a general symmetric family of quartic criteria a unified development of algorithms for orthogonal and direct oblique rotation using arbitrary criteria from this family is given these algorithms represent fairly straightforward extensions of present methodology and appear to be the best methods currently available

fuzzy fractional differential equations under caputo katugampola fractional derivative approach abstract in this work an initial value problem of caputo katugampola ck fractional differential equations in fuzzy setting is considered and an idea of successive approximations under generalized lipschitz condition is used to prove the existence and uniqueness results of the solution to the given problem in order to obtain the above results some necessary comparison theorems in real valued differential equation under ck fractional derivative are established finally a new technique to find analytical solutions of ck fuzzy fractional differential equations by using the solutions of fuzzy integer order differential equations is proposed some examples illustrating the applications of our results are presented

rotation to simple loadings using component loss functions the orthogonal case component loss functions clfs are used to generalize the quartimax criterion for orthogonal rotation in factor analysis these replace the fourth powers of the factor loadings by an arbitrary function of the second powers criteria of this form were introduced by a number of authors primarily katz and rohlf 1974 and rozeboom 1991 but there has been essentially no follow up to this work a method so simple natural and general deserves to be investigated more completely a number of theoretical results are derived including the fact that any method using a concave clf will recover perfect simple structure whenever it exists and there are methods that will recover thurstone simple structure whenever it exists specific clfs are identified and it is shown how to compare these using standardized plots numerical examples are used to illustrate and compare clf and other methods sorted absolute loading plots are introduced to aid in comparing results and setting parameters for methods that require them

exploratory bi factor analysis the oblique case bi factor analysis is a form of confirmatory factor analysis originally introduced by holzinger and swineford psychometrika 47 41 54 1937 the bi factor model has a general factor a number of group factors and an explicit bi factor structure jennrich and bentler psychometrika 76 537 549 2011 introduced an exploratory form of bi factor analysis that does not require one to provide an explicit bi factor structure a priori they use exploratory factor analysis and a bifactor rotation criterion designed to produce a rotated loading matrix that has an approximate bi factor structure among other things this can be used as an aid in finding an explicit bi factor structure for use in a confirmatory bi factor analysis they considered only orthogonal rotation the purpose of this paper is to consider oblique rotation and to compare it to orthogonal rotation because there are many more oblique rotations of an initial loading matrix than orthogonal rotations one expects the oblique results to approximate a bi factor structure better than orthogonal rotations and this is indeed the case a surprising result arises when oblique bi factor rotation methods are applied to ideal data

direct schmid leiman transformations and rank deficient loadings matrices the schmid leiman s l psychometrika 22 53 61 1957 transformation is a popular method for conducting exploratory bifactor analysis that has been used in hundreds of studies of individual differences variables to perform a two level s l transformation it is generally believed that two separate factor analyses are required a first level analysis in which k obliquely rotated factors are extracted from an observed variable correlation matrix and a second level analysis in which a general factor is extracted from the correlations of the first level factors in this article i demonstrate that the s l loadings matrix is necessarily rank deficient i then show how this feature of the s l transformation can be used to obtain a direct s l solution from an unrotated first level factor structure next i reanalyze two examples from mansolf and reise multivar behav res 51 698 717 2016 to illustrate the utility of best fitting s l rotations when gauging the ability of hierarchical factor models to recover known bifactor structures finally i show how to compute direct bifactor solutions for non hierarchical bifactor structures an online supplement includes r code to reproduce all of the analyses that are reported in the article

metric spaces of fuzzy sets two classes of metrics are introduced for spaces of fuzzy sets their equivalence is discussed and basic properties established a characterisation of compact and locally compact subsets is given in terms of boundedness and p mean equileft continuity and the spaces shown to be locally compact complete and separable metric spaces

generalizations of the differentiability of fuzzy number valued functions with applications to fuzzy differential equations the usual concept of differentiability of fuzzy number valued functions has the following shortcoming if c is a fuzzy number and g a b r is an usual real valued function differentiable on x 0 a b with g x 0 0 then f x c g x is not differentiable on x 0 in this paper we introduce and study generalized concepts of differentiability of any order n n which solves this shortcoming newton leibnitz type formula is obtained and existence of the solutions of fuzzy differential equations involving generalized differentiability is studied also some concrete applications to partial and ordinary fuzzy differential equations with fuzzy input data of the form c g x are given

a generalization of hukuhara difference and division for interval and fuzzy arithmetic we propose a generalization of the hukuhara difference first the case of compact convex sets is examined then the results are applied to generalize the hukuhara difference of fuzzy numbers using their compact and convex level cuts finally a similar approach is suggested to attempt a generalization of division for real intervals and fuzzy numbers applications to solving interval and fuzzy linear equations and fuzzy differential equations are shown

newton algorithms for analytic rotation an implicit function approach in this paper implicit function based parameterizations for orthogonal and oblique rotation matrices are proposed the parameterizations are used to construct newton algorithms for minimizing differentiable rotation criteria applied to m factors and p variables the speed of the new algorithms is compared to that of existing algorithms and to that of newton algorithms based on alternative parameterizations several rotation criteria were examined and the algorithms were evaluated over a range of values for m initial guesses for newton algorithms were improved by subconvergence iterations of the gradient projection algorithm simulation results suggest that no one algorithm is fastest for minimizing all criteria for all values of m among competing algorithms the gradient projection algorithm alone was faster than the implicit function algorithm for minimizing a quartic criterion over oblique rotation matrices when m is large in all other conditions however the implicit function algorithms were competitive with or faster than the fastest existing algorithms the new algorithms showed the greatest advantage over other algorithms when minimizing a nonquartic component loss criterion

a generalization of takane s algorithm for dedicom an algorithm is described for fitting the dedicom model for the analysis of asymmetric data matrices this algorithm generalizes an algorithm suggested by takane in that it uses a damping parameter in the iterative process takane s algorithm does not always converge monotonically based on the generalized algorithm a modification of takane s algorithm is suggested such that this modified algorithm converges monotonically it is suggested to choose as starting configurations for the algorithm those configurations that yield closed form solutions in some special cases finally a sufficient condition is described for monotonic convergence of takane s original algorithm

determination of point group harmonics for arbitrary j by a projection method ii icosahedral group quantization along an axis of order 5 the method described in the previous paper to obtain cubic harmonics is applied to the similar problem for the icosahedral group the projection operators are expressed in terms of rotation matrix elements of r 0 and r 0 arctan 2 acting on subspaces invariant under d5

elementary fuzzy calculus abstract in 2 dubois and prade introduced the notion of fuzzy numbers and defined the basic operations of addition subtraction multiplication and division a slightly modified definition of fuzzy numbers was presented in 4 and in that paper a metric was defined for this family of fuzzy sets another less restrictive definition of fuzzy numbers can be found in 1 in the present paper we consider fuzzy numbers from a somewhat different perspective basically we shall view fuzzy numbers in a topological vector space setting using the customary vector space operations together with the metric given in 4 we will define differentiation and integration of fuzzy valued functions in ways that parallel closely the corresponding definitions for real differentiation and integration

on fuzzy solutions for heat equation based on generalized hukuhara differentiability abstract in this paper a fuzzy heat equation with fuzzy initial values is considered the concept of generalized hukuhara differentiation is interpreted thoroughly in the univariate and multivariate cases and also several properties for generalized hukuhara differentiability are obtained on the topics such as switching point the univariate and multivariate fuzzy chain rules fuzzy mean value theorem among others the objective of this paper is to prove the uniqueness of a solution for a fuzzy heat equation and show that a fuzzy heat equation can be modeled as two systems of fuzzy differential equations by considering the type of differentiability of solutions finally some examples show the behavior of the solutions obtained

calculation of primitive 6 j symbols the 6 j symbols of a group are independent of the subgroup chain chosen to define the basis states the authors present an improved algorithm for calculating the primitive 6 j symbols for a compact group with a faithful irrep by recursive building up using only the kronecker product rules and the general properties of 6 j previously one has sometimes needed to search for the useful equations by systematically trying all equations which involve unknown 6 j the authors show that the primitive 6 j may all be easily solved in terms of a subclass the core 6 j they discuss how the core 6 j can usually be solved proving that the method is complete for so3 the authors conjecture that the algorithm is complete for all groups

a joint treatment of varimax rotation and the problem of diagonalizing symmetric matrices simultaneously in the least squares sense the present paper contains a lemma which implies that varimax rotation can be interpreted as a special case of diagonalizing symmetric matrices as discussed in multidimensional scaling it is shown that the solution by de leeuw and pruzansky is essentially equivalent to the solution by kaiser necessary and sufficient conditions for maxima and minima are derived from first and second order partial derivatives a counter example by gebhardt is reformulated and examined in terms of these conditions it is concluded that kaiser s method or equivalently the method by de leeuw and pruzansky is the most attractive method currently available for the problem at hand

on the relationship between the higher order factor model and the hierarchical factor model the relationship between the higher order factor model and the hierarchical factor model is explored formally we show that the schmid leiman transformation produces constrained hierarchical factor solutions using a generalized schmid leiman transformation and its inverse we show that for any unconstrained hierarchical factor model there is an equivalent higher order factor model with direct effects loadings on the manifest variables from the higher order factors therefore the class of higher order factor models without direct effects of higher order factors is nested within the class of unconstrained hierarchical factor models in light of these formal results we discuss some implications for testing the higher order factor model and the issue of general factor an interesting aspect concerning the efficient fitting of the higher order factor model with direct effects is noted

3 d image reconstruction from averaged fourier transform magnitude by parameter estimation an object model and estimation procedure for three dimensional 3 d reconstruction of objects from measurements of the spherically averaged fourier transform magnitudes is described the motivating application is the 3 d reconstruction of viruses based on solution x ray scattering data the object model includes symmetry positivity and support constraints and has the form of a truncated orthonormal expansion and the parameters are estimated by maximum likelihood methods successful 3 d reconstructions based on synthetic and experimental measurements from cowpea mosaic virus are described

on the zeros of 3j coefficients polynomial degree versus recurrence order for 3j coefficients of su 2 two quantities the degree n and the order m are defined the first the polynomial degree n is well known as the number of terms in the polynomial part of the single sum expression of the 3j coefficient minus one the second the recurrence order m is new from an exhaustive computer search for non trivial zeros of 3j coefficients it follows that both quantities play an important role in classifying the zeros explicit formulae for all zeros of order 1 are obtained and several new infinite sequences of zeros related to pell equations are presented

analysis and numerical approximation of tempered fractional calculus of variations problems abstract in this paper we study variational problems where the cost functional involves the tempered caputo fractional derivative several important optimization conditions are derived to find the optimal solution sufficient and necessary conditions are presented for different variational problems for example the cases of integral isoperimetric problem and holonomic constraints are considered as well as problems with high order derivatives a numerical scheme is proposed to determine approximations of the solution and it is illustrated through some examples

bayesian exploratory factor analysis this paper develops and applies a bayesian approach to exploratory factor analysis that improves on ad hoc classical approaches our framework relies on dedicated factor models and simultaneously determines the number of factors the allocation of each measurement to a unique factor and the corresponding factor loadings classical identification criteria are applied and integrated into our bayesian procedure to generate models that are stable and clearly interpretable a monte carlo study confirms the validity of the approach the method is used to produce interpretable low dimensional aggregates from a high dimensional set of psychological measurements

derivative free gradient projection algorithms for rotation a simple modification substantially simplifies the use of the gradient projection gp rotation algorithms of jennrich 2001 2002 these algorithms require subroutines to compute the value and gradient of any specific rotation criterion of interest the gradient can be difficult to derive and program it is shown that using numerical gradients gives almost precisely the same results as using exact gradients the resulting algorithm is very easy to use because the only problem specific code required is that needed to define the rotation criterion the computing time is increased when using numerical gradients but it is still very modest for most purposes while used extensively elsewhere numerical derivatives seem to be underutilized in statistics

identification with deficient rank loading matrices in confirmatory factor analysis multitrait multimethod models this paper presents some results on identification in multitrait multimethod mtmm confirmatory factor analysis cfa models some mtmm models are not identified when the factorial patterned loadings matrix is of deficient column rank for at least one other mtmm model identification does exist despite such deficiency it is also shown that for some mtmm cfa models howe s 1955 conditions sufficient for rotational uniqueness can fail yet the model may well be identified and rotationally unique implications of these results for cfa models in general are discussed

hyers ulam stability of fuzzy fractional volterra integral equations with the kernel ψ function via successive approximation method abstract in this paper we investigative the hyers ulam stability and the hyers ulam rassias stability of fuzzy fractional volterra integral equations ffvies involving the kernel function we propose conditions for the existence uniqueness and stability of ffvie by employing the method of successive approximation to visualize the theoretical results some examples and numerical simulations are given

necessary optimality conditions for fuzzy variational problems the main aims of this study are to derive the fuzzy euler lagrange conditions for both fuzzy unconstrained and constrained variational problems based on the concepts of differentiability and integrability of a fuzzy mapping that may be parameterized by the left and right hand functions of its a level sets

on a labeling for point group harmonics i cubic group expressions for the 4 6 and 8 representations are derived in terms of the 1 representations for neighboring angular momenta expressions for 5 7 and again for 8 are derived in terms of the 2 representations a third set of expressions for 8 is derived in terms of the 3 representations with an arbitrary choice of orthonormal sets of the 1 2 and 3 representations orthonormal sets of all other kinds of representations are thus well defined and can be entirely labeled using the parent representations all clebsch gordan coefficients are expressed in terms of those between parent representations and a few other ones the 4 and 5 representations defined here are not those conventionally used but they provide a simpler expression for the fictitious spin coefficients tables of the parent representations 1 2 and 3 are given for quaternary and ternary axes of quantization using the usual clebsch gordan coefficients of su 2 these tables allow us to obtain any representation for int

a simple general method for oblique rotation a simple and very general algorithm for oblique rotation is identified while motivated by the rotation problem in factor analysis it may be used to minimize almost any function of a not necessarily square matrix whose columns are restricted to have unit length the algorithm has two steps the first is to compute the gradient of the rotation criterion and the second is to project this onto a manifold of matrices with unit length columns for this reason it is called a gradient projection algorithm because the projection step is very simple implementation of the algorithm involves little more than computing the gradient of the rotation criterion which for many applications is very simple it is proven that the algorithm is strictly monotone that is as long as it is not already at a stationary point each step will decrease the value of the criterion examples from a variety of areas are used to demonstrate the algorithm including oblimin rotation target rotation simplimax rotation and rotation to similarity and simplicity simultaneously while it may be the algorithm is not intended for use as a standard algorithm for well established problems but rather as a tool for investigating new methods where its generality and simplicity may save an investigator substantial effort

symmetry constrained 3 d interpolation of viral x ray crystallography data a three dimensional 3 d interpolation problem that is important in viral x ray crystallography is considered the problem requires new methods because the function is known to have icosahedral symmetry the data is corrupted by experimental errors and therefore lacks the symmetry the problem is 3 d the measurements are irregularly spaced and the number of measurements is large 10 sup 4 a least squares approach is taken using two sets of basis functions the functions implied by a minimum energy bandlimited exact interpolation problem and a complete orthonormal set of bandlimited functions a numerical example of the cowpea mosaic virus is described

techniques for rotating two or more loading matrices to optimal agreement and simple structure a comparison and some technical details matrices of factor loadings are often rotated to simple structure when more than one loading matrix is available for the same variables the loading matrices can be compared after rotating them all separately to simple structure an alternative procedure is to rotate them to optimal agreement and then compare them in the present paper techniques are described that combine these two procedures specifically five techniques that combine the ideals of rotation to optimal agreement and rotation to simple structure are compared on the basis of contrived and empirical data for the contrived data it is assessed to what extent the rotations recover the underlying common structure for both the contrived and the empirical data it is studied to what extent the techniques give well matching rotated matrices to what extent these have a simple structure and to what extent the most prominent parts of the different loading matrices agree it was found that the simple procedure of combining a generalized procrustes analysis gpa with varimax on the mean of the matched loading matrices performs very well on all criteria and for most purposes offers an attractive compromise of rotation to agreement and simple structure in addition to this comparison some technical improvements are proposed for bloxom s rotation to simple structure and maximum similarity

a simple general procedure for orthogonal rotation a very general algorithm for orthogonal rotation is identified it is shown that when an algorithm parameter is sufficiently large the algorithm converges monotonically to a stationary point of the rotation criterion from any starting value because a sufficiently large is in general hard to find a modification that does not require it is introduced without this requirement the modified algorithm is not only very general but also very simple its implementation involves little more than computing the gradient of the rotation criterion while the modified algorithm converges monotonically from any starting value it is not guaranteed to converge to a stationary point it however does so in all of our examples while motivated by the rotation problem in factor analysis the algorithms discussed may be used to optimize almost any function of a not necessarily square column wise orthonormal matrix a number of these more general applications are considered empirical examples show that the modified algorithm can be reasonably fast but its purpose is to save an investigator s effort rather than that of his or her computer this makes it more appropriate as a research tool than as an algorithm for established methods

algebraic expressions for some so n racah coefficients in this paper the algebraic expressions for so n racah coefficients involving the coupling of three irreducible representations 1 0 over dot m 1 m 2 m 3 0 over dot are given by using the building up process their properties are discussed in detail c 1998 american institute of physics s0022 2488 98 01811 8

acceleration of the em algorithm by using quasi newton methods the em algorithm is a popular method for maximum likelihood estimation its simplicity in many applications and desirable convergence properties make it very attractive its sometimes slow convergence however has prompted researchers to propose methods to accelerate it we review these methods classifying them into three groups pure hybrid and em type accelerators we propose a new pure and a new hybrid accelerator both based on quasi newton methods and numerically compare these and two other quasi newton accelerators for this we use examples in each of three areas poisson mixtures the estimation of covariance from incomplete data and multivariate normal mixtures in these comparisons the new hybrid accelerator was fastest on most of the examples and often dramatically so in some cases it accelerated the em algorithm by factors of over 100 the new pure accelerator is very simple to implement and competed well with the other accelerators it accelerated the em algorithm in some cases by factors of over 50 to obtain standard errors we propose to approximate the inverse of the observed information matrix by using auxiliary output from the new hybrid accelerator a numerical evaluation of these approximations indicates that they may be useful at least for exploratory purposes

sparse versus simple structure loadings the component loadings are interpreted by considering their magnitudes which indicates how strongly each of the original variables relates to the corresponding principal component the usual ad hoc practice in the interpretation process is to ignore the variables with small absolute loadings or set to zero loadings smaller than some threshold value this in fact makes the component loadings sparse in an artificial and a subjective way we propose a new alternative approach which produces sparse loadings in an optimal way the introduced approach is illustrated on two well known data sets and compared to the existing rotation methods

faint phase synchronization detection through structured orthomax rotations in singular spectrum analysis multivariate singular spectrum analysis m ssa with a structured varimax rotation is a method that allows a deep characterization of phase synchronization ps phenomena in an almost automatic fashion it has been increasingly used in the study of ps in networks of nonlinear real world and numeric systems this paper investigates the impact of the other recently developed structured orthomax rotations on the m ssa ability to characterize ps the results show that by using the structured quartimax rotation a very faint and intermittent ps regime can be detected in contrast with the structured varimax which demands a stronger more consolidated ps regime this is due to the fact that the different rotations do not have the same efficiency in achieving a simple structure of the m ssa eigenvectors nevertheless for well established ps regimes the same robustness of the original m ssa approach against high levels of additive gaussian noise was found for the structured quartimax and biquartimax rotations however for all approaches we found an overshoot of the qualitative range for the ps onset due to noise

fuzzy differential equations abstract this paper deals with fuzzy set valued mappings of a real variable whose values are normal convex upper semicontinuous and compactly supported fuzzy sets in rn we study differentiability and integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy differential equation

predicting monthly streamflow using data driven models coupled with data preprocessing techniques 1 in this paper the accuracy performance of monthly streamflow forecasts is discussed when using data driven modeling techniques on the streamflow series a crisp distributed support vectors regression cdsvr model was proposed for monthly streamflow prediction in comparison with four other models autoregressive moving average arma k nearest neighbors knn artificial neural networks anns and crisp distributed artificial neural networks cdann with respect to distributed models of cdsvr and cdann the fuzzy c means fcm clustering technique first split the flow data into three subsets low medium and high levels according to the magnitudes of the data and then three single svrs or anns were fitted to three subsets this paper gives a detailed analysis on reconstruction of dynamics that was used to identify the configuration of all models except for arma to improve the model performance the data preprocessing techniques of singular spectrum analysis ssa and or moving average ma were coupled with all five models some discussions were presented 1 on the number of neighbors in knn 2 on the configuration of ann and 3 on the investigation of effects of ma and ssa two streamflow series from different locations in china xiangjiaba and danjiangkou were applied for the analysis of forecasting forecasts were conducted at four different horizons 1 3 6 and 1 2 month ahead forecasts the results showed that models fed by preprocessed data performed better than models fed by original data and cdsvr outperformed other models except for at a 6 month ahead horizon for danjiangkou for the perspective of streamflow series the ssa exhibited better effects on danjingkou data because its raw discharge series was more complex than the discharge of xiangjiaba the ma considerably improved the performance of ann cdann and cdsvr by adjusting the correlation relationship between input components and output of models it was also found that the performance of cdsvr deteriorated with the increase of the forecast horizon

a new image thresholding method based on gaussian mixture model in this paper an efficient approach to search for the global threshold of image using gaussian mixture model is proposed firstly a gray level histogram of an image is represented as a function of the frequencies of gray level then to fit the gaussian mixtures to the histogram of image the expectation maximization em algorithm is developed to estimate the number of gaussian mixture of such histograms and their corresponding parameterization finally the optimal threshold which is the average of these gaussian mixture means is chosen and the experimental results show that the new algorithm performs better

a survey on the initial value problems of fuzzy implicit fractional differential equations abstract the purpose of this paper is to present some basic theories of various classes of initial value problems of implicit fractional differential equation involving the caputo fuzzy type riemann liouville fuzzy type and hadamard fuzzy type concepts of fractional derivative in the case of the order 0 1 in particular the existence and uniqueness results of solution for the given initial value problems of implicit fractional differential equations is investigated in addition some results concerning continuous dependence of solutions on perturbations of the given parameters namely of the initial conditions of the function and parameters on the right hand side of the order of fractional derivative are analyzed finally some examples are given to illustrate our main results

factor analysis via components analysis when the factor analysis model holds component loadings are linear combinations of factor loadings and vice versa this interrelation permits us to define new optimization criteria and estimation methods for exploratory factor analysis although this article is primarily conceptual in nature an illustrative example and a small simulation show the methodology to be promising

towards fuzzy differential calculus part 3 differentiation this paper deals successively with differentiation of ordinary functions at a fuzzy point and with differentiation of fuzzy valued functions at a nonfuzzy point in both cases the concept of differentiation is extended and the problem of the linearity is addressed in the second case practical methods of computation are presented in practical situations the available knowledge about mappings may be fuzzy it is the motivation of this work

on a labeling for point group harmonics ii icosahedral group the expressions for the 2 5 6 8 and 9 representations are given in terms of those of 1 representations for neighboring angular momenta the coefficients of the 4 representations are expressed in terms of those of 7 representations therefore with an arbitrary choice of orthonormal sets of 1 3 and 7 representations orthonormal sets of other kinds of representations are well defined and can be labeled with the labels of parent representations all clebsch gordan coefficients are expressed in terms of those between parent representations and a few others tables of all nondegenerate 1 3 and 7 representations are given for the axes of quantization of order 5 with some degenerate 3 and 7 representations which are also given any representation of integer or half integer angular momentum up to 27 can be obtained using some usual clebsch gordan coefficients of su 2

fractional discrete time diffusion equation with uncertainty applications of fuzzy discrete fractional calculus abstract this study provides some basics of fuzzy discrete fractional calculus as well as applications to fuzzy fractional discrete time equations with theories of r cut set fuzzy caputo and riemann liouville fractional differences are defined on a isolated time scale discrete leibniz integral law is given by use of w monotonicity conditions furthermore equivalent fractional sum equations are established fuzzy discrete mittag leffler functions are obtained by the picard approximation finally fractional discrete time diffusion equations with uncertainty is investigated and exact solutions are expressed in form of two kinds of fuzzy discrete mittag leffler functions this paper suggests a discrete time tool for modeling discrete fractional systems with uncertainty

nonconservative lagrangian and hamiltonian mechanics traditional lagrangian and hamiltonian mechanics cannot be used with nonconservative forces such as friction a method is proposed that uses a lagrangian containing derivatives of fractional order a direct calculation gives an euler lagrange equation of motion for nonconservative forces conjugate momenta are defined and hamilton s equations are derived using generalized classical mechanics with fractional and higher order derivatives the method is applied to the case of a classical frictional force proportional to velocity textcopyright 1996 the american physical society

solving interval valued fractional initial value problems under caputo gh fractional differentiability abstract in this paper interval valued fractional differential equations ifdes under caputo generalized hukuhara differentiability are introduced we present existence and uniqueness results for ifdes with a krasnoselskii krein type condition the solution to interval valued fractional initial value problems under caputo type interval valued fractional derivatives by a modified fractional euler method mfem and a modified adams bashforth moulton method mabmm is also presented

fuzzy random variables abstract in this paper we define the concepts of fuzzy random variable and the expectation of a fuzzy random variable the new definition of expectation generalizes the integral of a set valued function we derive some properties of these new concepts by considering a suitable generalization of the hausdorff metric we derive the lebesgue dominated convergence type theorem

the improper fuzzy riemann integral and its numerical integration abstract the improper fuzzy riemann integral and its numerical integration are proposed in this paper under the setting of the improper fuzzy riemann integral we can find the expectation of fuzzy random variable the level set of the improper fuzzy riemann integral is a closed interval whose end points are the improper riemann integrals thus we provide a numerical method to approximate the improper fuzzy riemann integral by invoking the simpson s rule we also fit the end points closed interval of the level set of the improper fuzzy riemann integral as polynomials with variable in a least squares sense finally the membership function of the improper fuzzy riemann integral can be transformed into nonlinear programming problem and be solved by any presented optimizer