a continuum model accounting for defect and mass densities in solids with inelastic material behaviour in this paper the analysis of structures with inelastic material behaviour is considered taking into account the evolution of defects and changes in mass density the underlying kinematical concept of an oriented continuum is general enough to describe the micro and macrobehaviour of material bodies appropriately based on the logical and consistent variational arguments for a lagrangian functional the dynamic balance laws boundary and transversality conditions all related to the evolution of defect density and mass changes are derived for macro and microstresses of deformational as well as of configurational type the adopted procedure which formally leaves the balance laws unaltered leads to the additional balance law for changes in defect density and additional boundary conditions for the changes in mass and defect densities driving forces or affinities associated with the evolution of defect and mass densities and a generalization of the j integral representing the thermodynamic forces on defects are obtained a nonlocal constitutive model accounting for changes in the defect density is presented

conformal invariance in two dimensional percolation the word percolation borrowed from the latin refers to the seeping or oozing of a liquid through a porous medium usually to be strained in this and related senses it has been in use since the seventeenth century it was introduced more recently into mathematics by s r broadbent and j m hammersley bh and is a branch of probability theory that is especially close to statistical mechanics broadbent and hammersley distinguish between two types of spreading of a fluid through a medium or between two aspects of the probabilistic models of such processes diffusion processes in which the random mechanism is ascribed to the fluid and percolation processes in which it is ascribed to the medium a percolation process typically depends on one or more probabilistic parameters for example if molecules of a gas are absorbed at the surface of a porous solid as in a gas mask then their ability to penetrate the solid depends on the sizes of the pores in it and their positions both conceived to be distributed in some random manner a simple mathematical model of such a process is often defined by taking the pores to be distributed in some regular manner that could be determined by a periodic graph and to be open thus very large or closed thus smaller than the molecules with probabilities p and 1 p as p increases the probability of deeper penetration of the gas into the interior of the solid grows

on the universality of crossing probabilities in two dimensional percolation six percolation models in two dimensions are studied percolation by sites and by bonds on square hexagonal and triangular lattices rectangles of widtha and heightb are superimposed on the lattices and four functions representing the probabilities of certain crossings from one interval to another on the sides are measured numerically as functions of the ratioa b in the limits set by the sample size and by the conventions and on the range of the ratioa b measured the four functions coincide for the six models we conclude that the values of the four functions can be used as coordinates of the renormalization group fixed point

percolation transitions in two dimensions we investigate bond and site percolation models on several two dimensional lattices numerically by means of transfer matrix calculations and monte carlo simulations the lattices include the square triangular honeycomb kagome and diced lattices with nearest neighbor bonds and the square lattice with nearest and next nearest neighbor bonds results are presented for the bond percolation thresholds of the kagome and diced lattices and the site percolation thresholds of the square honeycomb and diced lattices we also include the bond and site percolation thresholds for the square lattice with nearest and next nearest neighbor bonds we find that corrections to scaling behave according to the second temperature dimension xt2 4 predicted by the coulomb gas theory and the theory of conformal invariance in several cases there is evidence for an additional term with the same exponent but modified by a logarithmic factor only for the site percolation problem on the triangular lattice does such a logarithmic term appear to be small or absent the amplitude of the power law correction associated with xt2 4 is found to be dependent on the orientation of the lattice with respect to the cylindrical geometry of the finite systems

tree graph inequalities and critical behavior in percolation models various inequalities are derived and used for the study of the critical behavior in independent percolation models in particular we consider the critical exponent associated with the expected cluster sizex and the structure of then site connection probabilities n x1 xn it is shown that quite generally 1 the upper critical dimension above which attains the bethe lattice value 1 is characterized both in terms of the geometry of incipient clusters and a diagramatic convergence condition for homogeneousd dimensional lattices with x y o x y d 2 atp p c our criterion shows that 1 if 6 d 3 the connectivity functions n are generally bounded by tree diagrams which involve the two point function we conjecture that above the critical dimension the asymptotic behavior of n in the critical regime is actually given by such tree diagrams modified by a nonsingular vertex factor other results deal with the exponential decay of the cluster size distribution and the function 2 x y

percolation thresholds and fractal dimensions for square and cubic lattices with long range correlated defects we study long range power law correlated disorder on square and cubic lattices in particular we present high precision results for the percolation thresholds and the fractal dimension of the largest clusters as a function of the correlation strength the correlations are generated using a discrete version of the fourier filtering method we consider two different metrics to set the length scales over which the correlations decay showing that the percolation thresholds are highly sensitive to such system details by contrast we verify that the fractal dimension d f is a universal quantity and unaffected by the choice of metric we also show that for weak correlations its value coincides with that for the uncorrelated system in two dimensions we observe a clear increase of the fractal dimension with increasing correlation strength approaching d f 2 the onset of this change does not seem to be determined by the extended harris criterion

the scaling limit of self avoiding random walk in high dimensions the brydges spencer lace expansion is used to prove that the scaling limit of the finite dimensional distributions of self avoiding random walk in the d dimensional cubic lattice mathbb z d is gaussian if d is sufficiently large it is also shown that the critical exponent gamma for the number of self avoiding walks is equal to 1 if d is sufficiently large

on the upper critical dimension of lattice trees and lattice animals we give a rigorous proof of mean field critical behavior for the susceptibility 1 2 and the correlation length v 1 4 for models of lattice trees and lattice animals in two cases i for the usual model with trees or animals constructed from nearest neighbor bonds in sufficiently high dimensions and ii for a class of spread out or long range models in which trees and animals are constructed from bonds of various lengths above eight dimensions this provides further evidence that for these models the upper critical dimension is equal to eight the proof involves obtaining an infrared bound and showing that a certain square diagram is finite at the critical point and uses an expansion related to the lace expansion for the self avoiding walk

mean field critical behaviour for percolation in high dimensions the triangle condition for percolation states that sum limits x y tau 0 x tau 0 y cdot tau y 0 is finite at the critical point where x y is the probability that the sitesx andy are connected we use an expansion related to the lace expansion for a self avoiding walk to prove that the triangle condition is satisfied in two situations i for nearest neighbour independent bond percolation on thed dimensional hypercubic lattice ifd is sufficiently large and ii in more than six dimensions for a class of spread out models of independent bond percolation which are believed to be in the same universality class as the nearest neighbour model the class of models in ii includes the case where the bond occupation probability is constant for bonds of length less than some large number and is zero otherwise in the course of the proof an infrared bound is obtained the triangle condition is known to imply that various critical exponents take their mean field bethe lattice values gamma beta 1 delta delta t 2 t underset raise0 3em hbox smash scriptscriptstyle geqslant 2 and that the percolation density is continuous at the critical point we also prove thatv2 in i and ii wherev2 is the critical exponent for the correlation length

exact enumeration of self avoiding walks on critical percolation clusters in 2 7 dimensions we study self avoiding walks on critical percolation clusters by means of a recently developed exact enumeration method which can handle walks of several thousand steps we had previously presented results for the two and three dimensional cases here we take a wider perspective and vary the system s dimensions up to d 7 beyond the supposed upper critical dimension of these results may serve to check analytical predictions and help understand how the medium s fractal structure impacts on the walks scaling behavior

local influence analysis of nonlinear structural equation models by regarding the latent random vectors as hypothetical missing data and based on the conditional expectation of the complete data log likelihood function in the em algorithm we investigate assessment of local influence of various perturbation schemes in a nonlinear structural equation model the basic building blocks of local influence analysis are computed via observations of the latent variables generated by the metropolis hastings algorithm while the diagnostic measures are obtained via the conformal normal curvature seven perturbation schemes including some perturbation schemes on latent vectors are investigated the proposed procedure is illustrated by a simulation study and a real example

assessing local influence for specific restricted likelihood application to factor analysis in restricted statistical models since the first derivatives of the likelihood displacement are often nonzero the commonly adopted formulation for local influence analysis is not appropriate however there are two kinds of model restrictions in which the first derivatives of the likelihood displacement are still zero general formulas for assessing local influence under these restrictions are derived and applied to factor analysis as the usually used restriction in factor analysis satisfies the conditions various influence schemes are introduced and a comparison to the influence function approach is discussed it is also shown that local influence for factor analysis is invariant to the scale of the data and is independent of the rotation of the factor loadings

uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation for independent translation invariant irreducible percolation models it is proved that the infinite cluster when it exists must be unique the proof is based on the convexity or almost convexity and differentiability of the mean number of clusters per site which is the percolation analogue of the free energy the analysis applies to both site and bond models in arbitrary dimension including long range bond percolation in particular uniqueness is valid at the critical point of one dimensional 1 x y 2 models in spite of the discontinuity of the percolation density there corollaries of uniqueness and its proof are continuity of the connectivity functions and except possibly at the critical point of the percolation density related to differentiability of the free energy are inequalities which bound the specific heat critical exponent in terms of the mean cluster size exponent and the critical cluster size distribution exponent e g 1 2 1 1

percolation on two and three dimensional lattices in this work we apply a highly efficient monte carlo algorithm recently proposed by newman and ziff to treat percolation problems the site and bond percolations are studied on a number of lattices in two and three dimensions quite good results for the wrapping probabilities correlation length critical exponent and critical concentration are obtained for the square simple cubic hexagonal close packed and hexagonal lattices by using relatively small systems we also confirm the universal aspect of the wrapping probabilities regarding site and bond dilution

critical percolation clusters in seven dimensions and on a complete graph we study critical bond percolation on a seven dimensional hypercubic lattice with periodic boundary conditions 7d and on the complete graph cg of finite volume number of vertices v we numerically confirm that for both cases the critical number density n s v of clusters of size s obeys a scaling form n s v s n over s v d f with identical volume fractal dimension d f 2 3 and exponent 1 1 d f 5 2 we then classify occupied bonds into bridge bonds which includes branch and junction bonds and nonbridge bonds a bridge bond is a branch bond if and only if its deletion produces at least one tree deleting branch bonds from percolation configurations produces leaf free configurations whereas deleting all bridge bonds leads to bridge free configurations composed of blobs it is shown that the fraction of nonbridge biconnected bonds vanishes n cg 0 for large cgs but converges to a finite value n 7d 0 0061931 7 for the 7d hypercube further we observe that while the bridge free dimension d bf 1 3 holds for both the cg and 7d cases the volume fractal dimensions of the leaf free clusters are different d lf 7d 0 669 9 2 3 and d lf cg 0 3337 17 1 3 on the cg and in 7d the whole leaf free and bridge free clusters all have the shortest path volume fractal dimension d min 1 3 characterizing their graph diameters we also study the behavior of the number and the size distribution of leaf free and bridge free clusters for the number of clusters we numerically find the number of leaf free and bridge free clusters on the cg scale as lnv while for 7d they scale as v for the size distribution we find the behavior on the cg is governed by a modified fisher exponent 1 while for leaf free clusters in 7d it is governed by fisher exponent 5 2 the size distribution of bridge free clusters in 7d displays two scaling behavior with exponents 4 and 1 the probability distribution p c 1 v dc 1 of the largest cluster of size c 1 for whole percolation configurations is observed to follow a single variable function p over x dx with x c 1 v d f for both cg and 7d up to a rescaling factor for the variable x the probability functions for cg and 7d collapse on top of each other within the entire range of x the analytical expressions in the x 0 and x limits are further confirmed our work demonstrates that the geometric structure of high dimensional percolation clusters cannot be fully accounted for by their complete graph counterparts

a note on the critical dimension for weakly self avoiding walks we consider a weakly self avoiding t step random walk on zdwith possible nonnearest neighbor jumps we prove that for d 3 the scaling limit of the end point is cauchy distributed as t

on the critical percolation probabilities we prove that the critical probabilities of site percolation on the square lattice satisfy the relation p c p 1 furthermore we prove the continuity of the function percolation probability

discriminative feature extraction by a neural implementation of canonical correlation analysis the canonical correlation analysis cca aims at measuring linear relationships between two sets of variables views that can be used for feature extraction in classification problems with multiview data however the correlated features extracted by the cca may not be class discriminative since cca does not utilize the class labels in its traditional formulation although there is a method called discriminative cca dcca that aims to increase the discriminative ability of cca inspired from the linear discriminant analysis lda it has been shown that the extracted features with this method are identical to those by the lda with respect to an orthogonal transformation therefore dcca is simply equivalent to applying single view regular lda to each one of the views separately besides dcca and the other similar dcca approaches have generalization problems due to the sample covariance matrices used in their computation which are sensitive to outliers and noisy samples in this paper we propose a method called discriminative alternating regression d ar to explore correlated and also discriminative features d ar utilizes two alternating multilayer perceptrons each with a linear hidden layer learning to predict both the class labels and the outputs of each other we show that the features found by d ar on training sets significantly accomplish higher classification accuracies on test sets of facial expression recognition object recognition and image retrieval experimental data sets

on local influence analysis of full information item factor models the full information item factor fiif model is very useful for analyzing relations of dichotomous variables in this article we present a feasible procedure to assess local influence of minor perturbations for identifying influence aspects of the fiif model the development is based on a q displacement function which is closely related with the monte carlo em algorithm in the ml estimation in the e step of this algorithm the conditional expectations are approximated by sample means of observations simulated by the gibbs sampler from the appropriate conditional distributions it turns out that these observations can be utilized for computing the building blocks of the proposed diagnostic measures the diagnoses are based on the conformal normal curvature that can be computed easily a number of interesting perturbation schemes are considered the methodology is illustrated with two real examples

critical percolation on the torus we compute the various crossing probabilities defined by r langlands p pouliot and y saint aubin for the critical percolation on the torus

bond and site percolation in three dimensions we simulate the bond and site percolation models on a simple cubic lattice with linear sizes up to l 512 and estimate the percolation thresholds to be p c mathrm bond 0 248 phantom rule 0 16em 0ex 811 phantom rule 0 16em 0ex 82 10 and p c mathrm site 0 311 phantom rule 0 16em 0ex 607 phantom rule 0 16em 0ex 7 2 by performing extensive simulations at these estimated critical points we then estimate the critical exponents 1 ensuremath nu 1 1410 15 ensuremath beta ensuremath nu 0 477 phantom rule 0 16em 0ex 05 15 the leading correction exponent y i ensuremath 1 2 2 and the shortest path exponent d mathrm min 1 3756 3 various universal amplitudes are also obtained including wrapping probabilities ratios associated with the cluster size distribution and the excess cluster number we observe that the leading finite size corrections in certain wrapping probabilities are governed by an exponent ensuremath approx ensuremath 2 rather than y i ensuremath approx ensuremath 1 2

influence in canonical correlation analysis the perturbation theory of the generalized eigenproblem is used to derive influence functions of each squared canonical correlation coefficient and the corresponding canonical vector pair three sample versions of these functions are described and some properties are noted as particular applications the influence function of the squared multiple correlation coefficient and influence functions of eigenvalues and eigenvectors in correspondence analysis are obtained three numerical examples are briefly discussed

sensitivity analysis of structural equation models the main purpose of this paper is to investigate the sensitivity analysis of structural equation model when minor perturbation is introduced some influence measure that based on the general case weight perturbation is derived for the generalized least squares estimation an influence measure that related to the cook s distance is also developed for the special case deletion perturbation scheme using the proposed methodology the influential observation in a data set can be detected moreover the general theory can be applied to detect the influential parameters in a model finally some illustrative artificial and real examples are presented

precise determination of the bond percolation thresholds and finite size scaling corrections for the sc fcc and bcc lattices extensive monte carlo simulations were performed to study bond percolation on the simple cubic s c face centered cubic f c c and body centered cubic b c c lattices using an epidemic kind of approach these simulations provide very precise values of the critical thresholds for each of the lattices pc s c 0 2488126 0 0000005 pc f c c 0 1201635 0 0000010 and pc b c c 0 1802875 0 0000010 for p close to pc the results follow the expected finite size and scaling behavior with values for the fisher exponent 2 189 0 002 the finite size correction exponent 0 64 0 02 and the scaling function exponent 0 445 0 01 confirmed to be universal

influential observations in principal factor analysis we propose a method for detecting influential observations in iterative principal factor analysis for this purpose we derive the influence functionsi x llt andi x for the common variance matrixt llt and the unique variance matrix respectively in the common factor decomposition llt a numerical example is given for illustration

phase diagram of inhomogeneous percolation with a defect plane let mathbb l be the d dimensional hypercubic lattice and let mathbb l 0 be an s dimensional sublattice with 2 le s p c d and a surface supercritical regime for p sigma p we show that sigma p is a strictly decreasing function for p in 0 p c d with a jump discontinuity at p c d we extend the aizenman barsky differential inequalities for homogeneous percolation to the inhomogeneous model and use them to prove that the susceptibility is finite inside the subcritical phase we prove that the cluster size distribution decays exponentially in the subcritical phase and sub exponentially in the supercritical phases for a model of lattice animals with a defect plane the free energy is related to functions of the inhomogeneous percolation model and we show how the percolation transition implies a non analyticity in the free energy of the animal model finally we present simulation estimates of the critical curve sigma p

strain measures integrability condition and frame indifference in the theory of oriented media this paper is concerned with the appropriate choice of state variables within the continuum model or oriented media it is shown that residual deformation strain and wryness can be considered as such quantities the compatibility conditions for them are derived which make the inverse problem of determining the displacement and director triad fields well posed the principle of frame indifference justifies the use of these quantities as state variables in the free energy density governing and constitutive equations are studied in detail a comparison with the continuum model of crystals with continuously distributed dislocations is provided

universality of 3d percolation exponents and first order corrections to scaling for conductivity exponents abstract by using a fast nested dissection algorithm we compare the results of finite size scaling at p c and of p scaling l const on large cubic random resistor networks up to 500 500 500 the p scaling for conductivity of both site and bond networks leads to an exponent t 2 00 1 the finite size scaling leads to the ratio of this conductivity exponent to the coherence length exponent t 2 283 3 combining these results we estimate 0 876 6 in excellent agreement with a value proposed by ballesteros et al the first order correctional exponent is found to be 1 0 2

the second order approximation to sample influence curve in canonical correlation analysis a second order approximation to the sample influence curve sic in canonical correlation analysis has been derived in the literature however it does not seem satisfactory for some cases in this paper we present a more accurate second order approximation as a particular case the proposed method is exact for the sic of the squared multiple correlation coefficient an example is given

direct estimates on intersection probabilities of random walks a study is made of intersection properties of independent random walks in d dimensional lattice space z sup d a simple method is developed which makes it possible to estimate intersection probabilities of two random walks with killing rate m directly it is expected that the method can be generalized and extended to other issues

on configurational forces in multiplicative elastoplasticity abstract the main goal of this work consists in the elaboration of the material or rather configurational mechanics in the context of multiplicative elastoplasticity this nowadays well established approach which is inherently related to the concept of a material isomorphism or in other words to a local rearrangement is adopted as a paradigm for the general modelling of finite inelasticity the overall motion in space is throughout assumed to be compatible and sufficiently smooth according to the underlying configurations namely the material and the spatial configuration as well as what we call the intermediate configuration different representations of balance of linear momentum are set up for the static case the underlying flux terms are thereby identified as stress tensors of piola and cauchy type and are assumed to derive from a free energy density function thus taking hyperelastic formats moreover the incorporated source terms namely the configurational volume forces are identified by comparison arguments these quantities include gradients of distortions as well as dislocation density tensors in particular those dislocation density tensors related to the elastic or plastic distortion do not vanish due to the general incompatibility of the intermediate configuration as a result configurational volume forces which are settled in the intermediate configuration embody non vanishing dislocation density tensors while their material counterparts directly incorporate non vanishing gradients of distortions this fundamental property enables us to recover the celebrated peach koehler force for finite inelasticity acting on a single dislocation from the intermediate configuration volume forces

on the determination of the crystal reference in nonlinear continuum theory of dislocations the paper is concerned with the determination of the anholonomic crystal reference from the given elastic strain metric and dislocation density cartan torsion the incompatible inverse elastic deformation is shown to satisfy a system of linear first order differential equations the integrability condition requires that the curvature of the crystal connection should vanish the equivalent system of equations and integrability condition for the inverse elastic rotation are also obtained formulae for the inverse elastic deformation and the inverse elastic rotation in form of tensor series are derived the dual problem of determining the plastic deformation and the plastic rotation from the plastic strain and the dislocation density is solved in a similar manner a linearization procedure is carried out and a comparison with kroner s theory is provided

gap exponents for percolation processes with triangle condition a study is made of the gap exponents for percolation processes with the triangle condition in the subcritical region it is show that the gaps are given by t 2 fort 2 3 scaling theory predicts thatp p c 0 s p p c p ande p 1 c 0 c 0 s p p c p 3 wheres p is the typical cluster size it is found that p c p p p c 0 s p 1 p c p 1 2 and p c p 3 e p 1 c 0 c 0 s p 1 p c p 3 4

spanning forests and osp n 2m invariant σ models the present paper is part of our ongoing work on osp n 2m supersymmetric models their relation with the potts model at q 0 and spanning forests and the rigorous analytic continuation of the partition function as an entire function of n 2m a feature first predicted by parisi and sourlas in the 1970s here we accomplish two main steps first we analyze in detail the role of the ising variables that arise when the constraint in the osp 1 2 model is solved and we point out two situations in which the ising and forest variables decouple second we establish the analytic continuation for the osp n 2m model in some special cases when the underlying graph is a forest and for the nienhuis action on a cubic graph we also make progress in understanding the series parallel graphs

on natural strain measures of the non linear micropolar continuum we discuss three different ways of defining the strain measures in the non linear micropolar continuum a by a direct geometric approach b considering the strain measures as the fields required by the structure of local equilibrium conditions and c requiring the strain energy density of the polar elastic body to satisfy the principle of invariance under superposed rigid body deformations the geometric approach a generates several two point deformation measures as well as some lagrangian and eulerian strain measures the ways b and c allow one to choose those lagrangian strain measures which satisfy the additional mechanical requirements these uniquely selected relative strain measures are called the natural ones all the strain measures discussed here are formulated in the general coordinate free form they are valid for unrestricted translations stretches and changes of orientations of the micropolar body and are required to identically vanish in the absence of deformation the relation of the lagrangian stretch and wryness tensors derived here to the ones proposed in the literature is thoroughly discussed

convergence of self avoiding random walk to brownian motion in high dimensions it is proved that in sufficiently high dimensions the scaled self avoiding random walk on the hypercubic lattice converges in distribution to brownian motion convergence of the finite dimensional distributions was shown elsewhere in this paper tightness is shown

universality of the excess number of clusters and the crossing probability function in three dimensional percolation extensive monte carlo simulations were performed to evaluate the excess number of clusters and the crossing probability function for three dimensional percolation on the simple cubic s c face centred cubic f c c and body centred cubic b c c lattices systems l l l0 with l0 l were studied for both bond s c f c c b c c and site f c c percolation the excess number of clusters q b per unit length was confirmed to be a universal quantity with a value q b 0 412 likewise the critical crossing probability in the l0 direction with periodic boundary conditions in thell plane was found to follow a universal exponential decay as a function of r d l 0 l for large r simulations were also carried out to find new precise values of the critical thresholds for site percolation on the f c c and b c c lattices yielding pc f c c d 0 199 2365 0 000 0010 pc b c c d 0 245 9615 0 000 0010 we also report the value pc s c d 0 311 6080 0 000 0004 for site percolation

the helicoidal modeling in computational finite elasticity part i variational formulation the finite elasticity mechanics of continua capable of a polar description is formulated by an alternative modeling to keeping position and orientation as uncoupled fields the rototranslation between two material particles can be described by a single complex tensorial quantity which is recognized to be orthogonal its linearization gives the characteristic curvature and differential vectors underlying the helicoidal modeling in both the sense of the body geometric description and the evolution of a deforming body after due introduction to dual tensors and rototranslations the polar description of the continuum is addressed with particular care to mixed differentiations of the rototranslation field then a thorough variational framework is established for the most general polar continuum under hyperelasticity hypothesis and the three field two field and one field principles are drawn and linearized the proposed modeling is expected to be profitably exploited in non linear finite element analyses of solids undergoing finite displacements rotations and strains

hyperscaling inequalities for percolation a set of critical exponent inequalities for independent percolation which saturate under the hyperscaling hypothesis is proved one of the consequences of the inequalities is the lower bounddc 6 for the upper critical dimension the proof is based on a rigorous version of the finite size scaling argument which extends easily to other systems such as ising ferromagnets

exponential convergence to equilibrium for a class of random walk models we prove exponential convergence to equilibrium l 2 geometric ergodicity for a random walk with inward drift on a sub cayley rooted tree this randomwalk model generalizes a monte carlo algorithm for the self avoiding walk proposed by berretti and sokal if the number of vertices of leveln in the tree grows asc n n n 1 we prove that the autocorrelation time satisfies n 2 n 1

percolation in media with columnar disorder we study a generalization of site percolation on a simple cubic lattice where not only single sites are removed randomly but also entire parallel columns of sites we show that typical clusters near the percolation transition are very anisotropic with different scaling exponents for the sizes parallel and perpendicular to the columns below the critical point there is a griffiths phase where cluster size distributions and spanning probabilities in the direction parallel to the columns have power law tails with continuously varying non universal powers this region is very similar to the griffiths phase in subcritical directed percolation with frozen disorder in the preferred direction and the proof follows essentially the same arguments as in that case but in contrast to directed percolation in disordered media the number of active growth sites in a growing cluster at criticality shows a power law while the probability of a cluster to continue to grow shows logarithmic behavior

sensitivity analysis in factor analysis difference between using covariance and correlation matrices influence curves of some parameters under various methods of factor analysis have been given in the literature these influence curves depend on the influence curves for either the covariance or the correlation matrix used in the analysis the differences between the influence curves based on the covariance and the correlation matrices are derived in this paper simple formulas for the differences of the influence curves based on the two matrices for the unique variance matrix factor loadings and some other parameter are obtained under scale invariant estimation methods though the influence curves themselves are in complex forms

a unified approach to exploratory factor analysis with missing data nonnormal data and in the presence of outliers factor analysis is regularly used for analyzing survey data missing data data with outliers and consequently nonnormal data are very common for data obtained through questionnaires based on covariance matrix estimates for such nonstandard samples a unified approach for factor analysis is developed by generalizing the approach of maximum likelihood under constraints statistical properties of the estimates for factor loadings and error variances are obtained a rescaled bartlett corrected statistic is proposed for evaluating the number of factors equivariance and invariance of parameter estimates and their standard errors for canonical varimax and normalized varimax rotations are discussed numerical results illustrate the sensitivity of classical methods and advantages of the proposed procedures

shortest path fractal dimension for percolation in two and three dimensions we carry out a high precision monte carlo study of the shortest path fractal dimension d mathrm min for percolation in two and three dimensions using the leath alexandrowicz method which grows a cluster from an active seed site a variety of quantities are sampled as a function of the chemical distance including the number of activated sites a measure of the radius and the survival probability by finite size scaling we determine d mathrm min 1 130 phantom rule 0 16em 0ex 77 2 and 1 3756 6 in two and three dimensions respectively the result in two dimensions rules out the recently conjectured value d mathrm min 217 192 deng et al phys rev e 81 020102 r 2010

scaling corrections site percolation and ising model in three dimensions using finite size scaling techniques we obtain accurate results for critical quantities of the ising model and the site percolation in three dimensions we pay special attention to parametrizing the corrections to scaling which is necessary to bring the systematic errors below the statistical ones

monte carlo study of the site percolation model in two and three dimensions we investigate the site percolation problem on the square and simple cubic lattices by means of a monte carlo algorithm that in fact simulates systems with size l d 1 x infinity where l specifies the linear system size this algorithm can be regarded either as an extension of the hoshen kopelman method or as a special case of the transfer matrix monte carlo technique various quantities such as the magnetic correlation function are sampled in the finite directions of the above geometry simulations are arranged such that both bulk and surface quantities can be sampled on the square lattice we locate the percolation threshold at p c 0 592 746 5 4 and determine two universal quantities as q gbc 0 930 34 1 and q gsc 0 793 72 3 which are associated with bulk and surface correlations respectively these values agree well with the exact values 2 5 48 and 2 1 3 respectively which follow from conformal invariance on the simple cubic lattice we locate the percolation threshold at p c 0 311 607 7 4 we further determine the bulk thermal and magnetic exponents as y t 1 1437 6 and y h 2 5219 2 respectively and the surface magnetic exponent at the ordinary phase transition as y o hs 1 0248 3

the diffusion of self avoiding random walk in high dimensions we use the brydges spencer lace expansion to prove that the mean square displacement of at step strictly self avoiding random walk in thed dimensional square lattice is asymptotically of the formdt ast approaches infinity ifd is sufficiently large the diffusion constantd is greater than one

on the critical behavior of the magnetization in high dimensional ising models we derive rigorously general results on the critical behavior of the magnetization in ising models as a function of the temperature and the external field for the nearest neighbor models it is shown that ind 4 dimensions the magnetization is continuous att c and its critical exponents take the classical values 3 and 1 2 with possible logarithmic corrections atd 4 the continuity and other explicit bounds formally extend tod 3 1 2 other systems to which the results apply include long range models ind 1 dimension with 1 x y couplings for which 2 1 replacesd in the above summary the results are obtained by means of differential inequalities derived here using the random current representation which is discussed in detail for the case of a nonvanishing magnetic field

simple cubic random site percolation thresholds for neighborhoods containing fourth nearest neighbors in this paper random site percolation thresholds for a simple cubic sc lattice with site neighborhoods containing next next next nearest neighbors 4nn are evaluated with monte carlo simulations a recently proposed algorithm with low sampling for percolation thresholds estimation bastas et al arxiv 1411 5834 is implemented for the studies of the top bottom wrapping probability the obtained percolation thresholds are p c 4nn 0 31160 12 p c 4nn nn 0 15040 12 p c 4nn 2nn 0 15950 12 p c 4nn 3nn 0 20490 12 p c 4nn 2nn nn 0 11440 12 p c 4nn 3nn nn 0 11920 12 p c 4nn 3nn 2nn 0 11330 12 and p c 4nn 3nn 2nn nn 0 10000 12 where 3nn 2nn and nn stand for next next nearest neighbors next nearest neighbors and nearest neighbors respectively as an sc lattice with 4nn neighbors may be mapped onto two independent interpenetrated sc lattices but with a lattice constant that is twice as large the percolation threshold p c 4nn is exactly equal to p c nn the simplified method of bastas et al allows for uncertainty of the percolation threshold value p c to be reached similar to that obtained with the classical method but ten times faster

influence in principal components analysis summary in linear regression the theoretical influence function and the various sample versions of it have an established place as diagnostic tools these same functions are developed here to provide methods for the detection of influential observations in principal components analysis the perturbation theory of real symmetric matrices unifies this development some interesting points of contrast with the regression case are noted and explained theoretically

percolation critical exponents under the triangle condition for independent percolation models it is shown that if the diagrammatic triangle condition is satisfied then the critical exponents delta and hat beta exist and take their mean field values generalizing the criterion introduced in 1984 by aizenman and newman for the mean field value of gamma in nonoriented percolation the results apply to a broad class of nonoriented as well as oriented weakly homogeneous models in which the range of the connecting bonds need not be bounded for the nonoriented case the condition reduces to the finiteness at the critical point of nabla equiv sum x y tau 0 x tau x y tau y 0 with tau u nu the probability that the site u is connected to nu which was recently established by hara and slade for models with sufficiently spread out connections in d 6 dimensions our analysis proceeds through the derivation of complementary differential inequalities for the percolation order parameter m beta h whose value at h 0 yields the percolation density with beta parametrizing the bond or site occupation probabilities and with h h geq 0 a ghost field the conclusion is that under the triangle condition in the vicinity of the critical point beta c 0 m beta 0 approx beta beta c hat beta and m beta c h approx h 1 delta with hat beta 1 and delta 2

crossover from isotropic to directed percolation we generalize the directed percolation dp model by relaxing the strict directionality of dp such that propagation can occur in either direction but with anisotropic probabilities we denote the probabilities as p p pd and p p 1 pd with p representing the average occupation probability and pd controlling the anisotropy the leath alexandrowicz method is used to grow a cluster from an active seed site we call this model with two main growth directions biased directed percolation bdp standard isotropic percolation ip and dp are the two limiting cases of the bdp model corresponding to pd 1 2 and pd 0 1 respectively in this work besides ip and dp we also consider the 1 2 pd 1 region extensive monte carlo simulations are carried out on the square and the simple cubic lattices and the numerical data are analyzed by finite size scaling we locate the percolation thresholds of the bdp model for pd 0 6 and 0 8 and determine various critical exponents these exponents are found to be consistent with those for standard dp we also determine the renormalization exponent associated with the asymmetric perturbation due to pd 1 2 6 0 near ip and confirm that such an asymmetric scaling field is relevant at ip

sharpness of the phase transition in percolation models the equality of two critical points the percolation threshold ph and the point p where the cluster size distribution ceases to decay exponentially is proven for all translation invariant independent percolation models on homogeneous d dimensional lattices d 1 the analysis is based on a pair of new nonlinear partial differential inequalities for an order parameter m h which for h q reduces to the percolation density p at the bond density p l e in the single parameter case these are 1 m hdm dh m2 mdm d and 2 dm d j mdm dh inequality 1 is intriguing in that its derivation provides yet another hint of a 3 structure in percolation models moreover through the elimination of one of its derivatives 1 yields a pair of ordinary differential inequalities which provide information on the critical exponents and one of these resembles an ising model inequality of fr hlich and sokal and yields the mean field bound 5 2 and the other implies the result of chayes and chayes that an inequality identical to 2 is known for ising models where it provides the basis for newman s universal relation 5 1 1 and for certain extrapolation principles which are now made applicable also to independent percolation these results apply to both finite and long range models with or without orientation and extend to periodic and weakly inhomogeneous systems