approximation of the levy feller advection dispersion process by random walk and finite difference method in this paper we present a random walk model for approximating a levy feller advection dispersion process governed by the levy feller advection dispersion differential equation lfade we show that the random walk model converges to lfade by use of a properly scaled transition to vanishing space and time steps we propose an explicit finite difference approximation efda for lfade resulting from the grunwald letnikov discretization of fractional derivatives as a result of the interpretation of the random walk model the stability and convergence of efda for lfade in a bounded domain are discussed finally some numerical examples are presented to show the application of the present technique

new proofs of bapat and sivasubramanian s theorems abstract recently bapat and sivasubramanian 1 presented formulas for the determinant and the inverse of the product distance matrix of a tree furthermore they defined a bivariant ihara selberg zeta function of a graph and gave its determinant expression we present new proofs for three results of bapat and sivasubramanian

the robustness of estimates of total indirect effects in covariance structure models estimated by maximum likelihood the large sample distribution of total indirect effects in covariance structure models in well known using monte carlo methods this study examines the applicability of the large sample theory to maximum likelihood estimates oftotal indirect effects in sample sizes of 50 100 200 400 and 800 two models are studied model 1 is a recursive model with observable variables and model 2 is a nonrecursive model with latent variables for the large sample theory to apply the results suggest that sample szes of 200 or more and 400 or more are required for models such as model 1 and model 2 respectively

a new bartholdi zeta function of a digraph we introduce a new type of the bartholdi zeta function of a digraph d and give a determinant expression of it furthermore we define a new type of the bartholdi l function of d and give a determinant expression of it as a corollary we obtain a decomposition formula for a new type of the bartholdi zeta function of a group covering of d by new bartholdi l functions of d

a new weighted ihara zeta function for a graph abstract we define a new weighted ihara zeta function of a graph g and present its determinant expression we present a decomposition formula for the new weighted ihara zeta function of a regular covering of g furthermore we introduce a new weighted ihara l function of g and give determinant expressions of it as a corollary we present a decomposition formula for the new weighted ihara zeta function of a regular covering of g by its new weighted ihara l functions as applications we give new proofs for the results of kempton on the spectrum of the transition probability matrices for non backtracking random walks for regular graphs and semiregular bipartite graphs

numerical solution of the space fractional fokker planck equation the traditional second order fokker planck equation may not adequately describe the movement of solute in an aquifer because of large deviation from the dynamics of brownian motion densities of stable type have been used to describe the probability distribution of these motions the resulting governing equation of these motions is similar to the traditional fokker planck equation except that the order of the highest derivative is fractional in this paper a space fractional fokker planck equation sffpe with instantaneous source is considered a numerical scheme for solving sffpe is presented using the riemann liouville and grunwald letnikov definitions of fractional derivatives the sffpe is transformed into a system of ordinary differential equations ode then the ode system is solved by a method of lines numerical results for sffpe with a constant diffusion coefficient are evaluated for comparison with the known analytical solution the numerical approximation of sffpe with a time dependent diffusion coefficient is also used to simulate levy motion with stable densities we will show that the numerical method of sffpe is able to more accurately model these heavy tailed motions

zeta functions of quantum graphs in this paper we construct zeta functions of quantum graphs using a contour integral technique based on the argument principle we start by considering the special case of the star graph with neumann matching conditions at the center of the star we then extend the technique to allow any matching conditions at the center for which the laplace operator is self adjoint and finally obtain an expression for the zeta function of any graph with general vertex matching conditions in the process it is convenient to work with new forms for the secular equation of a quantum graph that extend the well known secular equation of the neumann star graph in the second half of this paper we apply the zeta function to obtain new results for the spectral determinant vacuum energy and heat kernel coefficients of quantum graphs these have all been topics of current research in their own right and in each case this unified approach significantly expands results in the literature

the coefficients of the reduced bartholdi zeta function abstract in this paper we establish a new zeta function based on the bartholdi zeta function for an undirected graph g called the reduced bartholdi zeta function we study the relation between its coefficients and the structure of the graph and demonstrate that the coefficients count the star subgraphs in the symmetric digraph d g moreover we investigate the properties of semi principle minors extracted from the adjacency matrix of the oriented line graph of g we also present a general formula for calculating all the coefficients of the reduced bartholdi zeta function

riemann roch and abel jacobi theory on a finite graph abstract it is well known that a finite graph can be viewed in many respects as a discrete analogue of a riemann surface in this paper we pursue this analogy further in the context of linear equivalence of divisors in particular we formulate and prove a graph theoretic analogue of the classical riemann roch theorem we also prove several results analogous to classical facts about riemann surfaces concerning the abel jacobi map from a graph to its jacobian as an application of our results we characterize the existence or non existence of a winning strategy for a certain chip firing game played on the vertices of a graph

the fractional diffusion equation in one space and in one time dimension a diffusion equation is solved where the first time derivative is replaced by the fractional time derivative 0 1 the solution is given in closed form in terms of fox functions

finite difference methods for two dimensional fractional dispersion equation fractional order partial differential equations as generalizations of classical integer order partial differential equations are increasingly used to model problems in fluid flow finance and other areas of application in this paper we discuss a practical alternating directions implicit method to solve a class of two dimensional initial boundary value fractional partial differential equations with variable coefficients on a finite domain first order consistency unconditional stability and therefore first order convergence of the method are proven using a novel shifted version of the classical grunwald finite difference approximation for the fractional derivatives a numerical example with known exact solution is also presented and the behavior of the error is examined to verify the order of convergence

the effect of sampling error on convergence improper solutions and goodness of fit indices for maximum likelihood confirmatory factor analysis a monte carlo study assessed the effect of sampling error and model characteristics on the occurrence of nonconvergent solutions improper solutions and the distribution of goodness of fit indices in maximum likelihood confirmatory factor analysis nonconvergent and improper solutions occurred more frequently for smaller sample sizes and for models with fewer indicators of each factor effects of practical significance due to sample size the number of indicators per factor and the number of factors were found for gfi agfi and rmr whereas no practical effects were found for the probability values associated with the chi square likelihood ratio test

the accuracy and stability of an implicit solution method for the fractional diffusion equation we have investigated the accuracy and stability of an implicit numerical scheme for solving the fractional diffusion equation this model equation governs the evolution for the probability density function that describes anomalously diffusing particles anomalous diffusion is ubiquitous in physical and biological systems where trapping and binding of particles can occur the implicit numerical scheme that we have investigated is based on finite difference approximations and is straightforward to implement the accuracy of the scheme is o dx 2 in the spatial grid size and o dt 1 c in the fractional time step where 0 email protected 1 is the order of the fractional derivative and c 1 is standard diffusion we have provided algebraic and numerical evidence that the scheme is unconditionally stable for 0 c 1

finite difference approximations for fractional advection dispersion flow equations fractional advection dispersion equations are used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium in this paper we develop practical numerical methods to solve one dimensional fractional advection dispersion equations with variable coefficients on a finite domain the practical application of these results is illustrated by modeling a radial flow problem use of the fractional derivative allows the model equations to capture the early arrival of tracer observed at a field site

improper solutions in the analysis of covariance structures their interpretability and a comparison of alternate respecifications a monte carlo approach was employed to investigate the interpretability of improper solutions caused by sampling error in maximum likelihood confirmatory factor analysis four models were studied with two sample sizes of the overall goodness of fit indices provided by the lisrel vi program significant differences between improper and proper solutions were found only for the root mean square residual as expected indicators of the factor on which the negative uniqueness estimate occurred had biased loadings and the correlations of its factor with other factors were also biased in contrast the loadings of indicators on other factors and those factor intercorrelations did not have any bias of practical significance for initial solutions with one negative uniqueness estimate three respecifications were studied fix the uniqueness at 00 fix it at 20 or constrain the domain of the solution to be proper for alternate respecified solutions that were converged and proper the constrained solutions and uniqueness fixed at 00 solutions were equivalent the mean goodness of fit and pattern coefficient values for the original improper solutions were not meaningfully different from those obtained under the constrained and uniqueness fixed at 00 respecifications

zeta functions of discrete groups acting on trees abstract this paper generalizes bass work on zeta functions for uniform tree lattices using the theory of von neumann algebras machinery is developed to define the zeta function of a discrete group of automorphisms of a bounded degree tree the main theorems relate the zeta function to determinants of operators defined on edges or vertices of the tree a zeta function associated to a non uniform tree lattice with appropriate hilbert representation is defined zeta functions are defined for infinite graphs with a cocompact or finite covolume group action

trace formulae and spectral statistics for discrete laplacians on regular graphs i following the derivation of the trace formulas in the first paper in this series we establish here a connection between the spectral statistics of random regular graphs and the predictions of random matrix theory rmt this follows from the known poisson distribution of cycle counts in regular graphs in the limit that the cycle periods are kept constant and the number of vertices increases indefinitely the result is analogous to the so called diagonal approximation in quantum chaos we also show that by assuming that the spectral correlations are given by rmt to all orders we can compute the leading deviations from the poisson distribution for cycle counts we provide numerical evidence which supports this conjecture

the asymptotic covariance matrix of maximum likelihood estimates in factor analysis the case of nearly singular matrix of estimates of unique variances this paper is concerned with the asymptotic covariance matrix acm of maximum likelihood estimates mles of factor loadings and unique variances when one element of mles of unique variances is nearly zero i e the matrix of mles of unique variances is nearly singular in this situation standard formulas break down we give explicit formulas for the acm of mles of factor loadings and unique variances that could be used even when an element of mles of unique variances is very close to zero we also discuss an alternative approach using the augmented information matrix under a nearly singular matrix of mles of unique variances and derive the partial derivatives of the alternative constraint functions with respect to the elements of factor loadings and unique variances 2000 elsevier science inc all rights reserved ams classification 62h25 62f12

the fractional fokker planck equation on comb like model abstract the fractional fokker planck equation were used to describe the anomalous diffusion in external fields is derived using a comb like structure as a background model for the force free case the distribution function associated with space dependence diffusion coefficient along the backbone of the structure are obtained in a closed form of h function the operator method has been used to solve the fractional fokker planck equation taking the external field into account

an explicit finite difference method and a new von neumann type stability analysis for fractional diffusion equations a numerical method for solving the fractional diffusion equation which could also be easily extended to other fractional partial differential equations is considered in this paper we combine the forward time centered space ftcs method well known for the numerical integration of ordinary diffusion equations with the grunwald letnikov discretization of the riemann liouville derivative to obtain an explicit ftcs scheme for solving the fractional diffusion equation the stability analysis of this scheme is carried out by means of a powerful and simple new procedure close to the well known von neumann method for nonfractional partial differential equations the analytical stability bounds are in excellent agreement with numerical test a comparison between exact analytical solutions and numerical predictions is made

the non backtracking spectrum of the universal cover of a graph a non backtracking walk on a graph h is a directed path of directed edges of h such that no edge is the inverse of its preceding edge non backtracking walks of a given length can be counted using the non backtracking adjacency matrix b indexed by h s directed edges and related to ihara s zeta function we show how to determine b s spectrum in the case where h is a tree covering a finite graph we show that when h is not regular this spectrum can have positive measure in the complex plane unlike the regular case we show that outside of b s spectrum the corresponding green function has periodic decay ratios the existence of such a ratio system can be effectively checked and is equivalent to being outside the spectrum we also prove that the spectral radius of the non backtracking walk operator on the tree covering a finite graph is exactly gr where gr is the growth rate of the tree this further motivates the definition of the graph theoretical riemann hypothesis proposed by stark and terras st finally we give experimental evidence that for a fixed finite graph h a ran dom lift of large degree has non backtracking new spectrum near that of h s universal cover this suggests a new generalization of alon s second eigenvalue conjecture

using phantom and imaginary latent variables to parameterize constraints in linear structural models the most widely used computer programs for structural equation models analysis are the lisrel series of joreskog and sorbom the only types of constraints which may be made directly are fixing parameters at a constant value and constraining parameters to be equal rindskopf 1983 showed how these simple properties could be used to represent models with more complicated constraints namely inequality constraints on unique variances in this paper two new concepts are introduced which enable a much wider variety of constraints to be made the concepts phantom and imaginary latent variables allow fairly general equality and inequality constraints on factor loadings and structural model coefficients

backtrackless walks on a graph the aim of this paper is to explore the use of backtrackless walks and prime cycles for characterizing both labeled and unlabeled graphs the reason for using backtrackless walks and prime cycles is that they avoid tottering and can increase the discriminative power of the resulting graph representation however the use of such methods is limited in practice because of their computational cost in this paper we present efficient methods for computing graph kernels which are based on backtrackless walks in a labeled graph and whose worst case running time is the same as that of kernels based on random walks for clustering unlabeled graphs we construct feature vectors using ihara coefficients since these coefficients are related to the frequencies of prime cycles in the graph to efficiently compute the low order coefficients we present an o v 3 algorithm which is better than the o v 6 worst case running time of previously known algorithms in the experimental evaluation we apply the proposed method to clustering both labeled and unlabeled graphs the results show that using backtrackless walks and prime cycles instead of random walks can increase the accuracy of recognition

zeta functions of several corona type graphs abstract in this paper we present the decomposition formulae for the zeta functions of families of graphs obtained through corona type construction furthermore we construct some families of graphs with the same zeta function by the spectra of these graphs

zeta functions of digraphs abstract we define a zeta function of a digraph and an l function of a symmetric digraph and give determinant expressions of them furthermore we give a decomposition formula for the zeta function of a g cyclic cover of a symmetric digraph for any finite group and g

a fourier method for the fractional diffusion equation describing sub diffusion in this paper a fractional partial differential equation fpde describing sub diffusion is considered an implicit difference approximation scheme idas for solving a fpde is presented we propose a fourier method for analyzing the stability and convergence of the idas derive the global accuracy of the idas and discuss the solvability finally numerical examples are given to compare with the exact solution for the order of convergence and simulate the fractional dynamical systems

nonconvergence improper solutions and starting values in lisrel maximum likelihood estimation in the framework of a robustness study on maximum likelihood estimation with lisrel three types of problems are dealt with nonconvergence improper solutions and choice of starting values the purpose of the paper is to illustrate why and to what extent these problems are of importance for users of lisrel the ways in which these issues may affect the design and conclusions of robustness research is also discussed

determinants of standard errors of mles in confirmatory factor analysis this paper studies changes of standard errors se of the normal distribution based maximum likelihood estimates mle for confirmatory factor models as model parameters vary using logical analysis simplified formulas and numerical verification monotonic relationships between ses and factor loadings as well as unique variances are found conditions under which monotonic relationships do not exist are also identified such functional relationships allow researchers to better understand the problem when significant factor loading estimates are expected but not obtained and vice versa what will affect the likelihood for heywood cases negative unique variance estimates is also explicit through these relationships empirical findings in the literature are discussed using the obtained results

implicit numerical approximation scheme for the fractional fokker planck equation in this paper we consider an anomalous subdiffusion process governed by fractional fokker planck equation an effective numerical method for approximating fokker planck equation in a bounded domain is presented the stability and convergence of the numerical method are analyzed some numerical examples are presented to show the application of the present technique the numerical results exhibit the good performance of our theoretical analysis

computational aspects of fem approximation of fractional advection dispersion equations on bounded domains in r 2 in this paper we investigate the numerical approximation of the variational solution to the fractional advection dispersion equation fade on bounded domains in r2 more specifically we investigate the computational aspects of the galerkin approximation using continuous piecewise polynomial basis functions on a regular triangulation of the domain the computational challenges of approximating the solution to fractional differential equations using the finite element method stem from the fact that a fractional differential operator is a non local operator several numerical examples are given which demonstrate approximations to fades

walk generating functions and spectral measures of infinite graphs let g be a locally finite graph with bounded vertex degrees its adjacency matrix a gives rise to a bounded self adjoint operator on l2 v g let e be the resolution of the identity for a and ev the vector in l2 v g having the v coordinate 1 and all other coordinates 0 the function uv e eu ev is called the spectral measure of g corresponding to vertices u v of g it is related to the generating function wuv z for the number of walks of given length between u and v the spectral measures are used to show the following the pairing theorem characterizing finite bipartite graphs in terms of their spectrum is generalized from finite to infinite graphs the average closed walk generating function can be defined in some cases generalizing the plancherel measure to not necessarily vertex transitive graphs some theorems about the convergence of graph measures are given with applications to the calculation of the expected eigenvalue distribution of a large finite random regular or bipartite semiregular graph and the calculation of spectral measures of several examples

some weighted bartholdi zeta function of a digraph related to the weighted bond scattering matrix of a graph we define some weighted bartholdi zeta function of a digraph d and give a determinant expression for it we present a decomposition formula for the weighted zeta function of a group covering of d furthermore we define an l function of d and give a determinant expression for it as a corollary we express the weighted bartholdi zeta function of a group covering of d by means of its l functions

new solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation a physical mathematical approach to anomalous diffusion is based on a generalized diffusion equation containing derivatives of fractional order in this paper an anomalous subdiffusion equation asub de is considered a new implicit numerical method inm and two solution techniques for improving the order of convergence of the inm for solving the asub de are proposed the stability and convergence of the inm are investigated by the energy method some numerical examples are given the numerical results demonstrate the effectiveness of theoretical analysis these methods and supporting theoretical results can also be applied to other fractional integro differential equations and higher dimensional problems

factor analysis with em algorithm never gives improper solutions when sample covariance and initial parameter matrices are proper rubin and thayer psychometrika 47 69 76 1982 proposed the em algorithm for exploratory and confirmatory maximum likelihood factor analysis in this paper we prove the following fact the em algorithm always gives a proper solution with positive unique variances and factor correlations with absolute values that do not exceed one when the covariance matrix to be analyzed and the initial matrices including unique variances and inter factor correlations are positive definite we further numerically demonstrate that the em algorithm yields proper solutions for the data which lead the prevailing gradient algorithms for factor analysis to produce improper solutions the numerical studies also show that in real computations with limited numerical precision rubin and thayer s psychometrika 47 69 76 1982 original formulas for confirmatory factor analysis can make factor correlation matrices asymmetric so that the em algorithm fails to converge however this problem can be overcome by using an em algorithm in which the original formulas are replaced by those guaranteeing the symmetry of factor correlation matrices or by formulas used to prove the above fact

zeta functions of finite graphs and coverings iii abstract a graph theoretical analog of brauer siegel theory for zeta functions of number fields is developed using the theory of artin l functions for galois coverings of graphs from parts i and ii in the process we discuss possible versions of the riemann hypothesis for the ihara zeta function of an irregular graph

kac ward formula and its extension to order disorder correlators through a graph zeta function a streamlined derivation of the kac ward formula for the planar ising model s partition function is presented and applied in relating the kernel of the kac ward matrices inverse with the correlation functions of the ising model s order disorder correlation functions a shortcut for both is facilitated by the bowen lanford graph zeta function relation the kac ward relation is also extended here to produce a family of non planar interactions on mathbb z 2 for which the partition function and the order disorder correlators are solvable at special values of the coupling parameters temperature

parameterizing inequality constraints on unique variances in linear structural models current computer programs for analyzing linear structural models will apparently handle only two types of constraints fixed parameters and equality of parameters an important constraint not handled is inequality this is particularly crucial for preventing negative variance estimates in this paper a method is described for imposing several kinds of inequality constraints in models without the necessity for having computer programs which explicitly allow such constraints the examples discussed include the prevention of heywood cases extension to inequalities of parameters to be greater than a specified value and imposing ordered inequalities

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems

simultaneous equation systems as moment structure models with an introduction to latent variable models abstract an introduction to latent variable covariance structure models is provided estimation and testing of such models with normal and non normal variables is reviewed and a relatively inexpensive asymptotically distribution free estimator is introduced the traditional econometric simultaneous equation system is reconceptualized as a random vector structural equation model extensions of the model to deal with latent variables and a wider variety of structural phenomena are made via the joreskog keesling wiley lisrel approach and a simpler approach due to bentler and weeks these representation systems are extended via first and second moments about the origin of measured variables explained in terms of the structural coefficients and the moments of explanatory variables citations to further literature are provided