field theory conjecture for loop erased random walks we give evidence that the functional renormalization group frg developed to study disordered systems may provide a field theoretic description for the loop erased random walk lerw allowing to compute its fractal dimension in a systematic expansion in e 4 d up to two loop the frg agrees with rigorous bounds correctly reproduces the leading logarithmic corrections at the upper critical dimension d 4 and compares well with numerical studies we obtain the universal subleading logarithmic correction in d 4 which can be used as a further test of the conjecture

on preconditioned iterative methods for burgers equations we study the newton method and a fixed point method for solving the system of nonlinear equations arising from the sinc galerkin discretization of the burgers equations in each step of the newton method or the fixed point method a structured subsystem of linear equations is obtained and needs to be solved numerically in this paper preconditioning techniques are applied to solve such linear subsystems the bounds for eigenvalues of the preconditioned matrices are derived and numerical examples are given to illustrate the effectiveness of the proposed methods we also find that a combination of the newton fixed point iteration with the preconditioned gmres method is quite efficient for the sinc galerkin discretization of the burgers equations

loop erased self avoiding random walk and the laplacian random walk comments that the laplacian random walk with eta 1 recently introduced by lyklema and evertsz 1986 is the same process as the loop erased self avoiding random walk analysed previously by the author 1980 some rigorous results about this process are reviewed

displacement rank of the drazin inverse in this paper we study the displacement rank of the drazin inverse both sylvester displacement and the generalized displacement are discussed we present upper bounds for the ranks of the displacements of the drazin inverse the general results are applied to the group inverse of a structured matrix such as close to toeplitz generalized cauchy toeplitz plus hankel and bezoutians

noise induced chaos and phase space flux abstract we study the effect of additive noise on near integrable second order dynamical systems whose unperturbed flows have homoclinic or heteroclinic orbits the noise is represented by a type of shinozuka stochastic process capable of arbitrarily closely approximating gaussian noise with any specified spectrum we derive a formula for the flux factor applicable for any asymptotic mean stationary excitation this derivation shows that to first order the effect of the external excitation on the system is mediated by a linear filter associated with the system homoclinic or heteroclinic orbit it also shows that the stationary mean distribution of the filtered excitation determines the average phase space flux this is true for both random and nonrandom excitations and indicates that for the dynamical systems considered here these two classes of excitation play substantially equivalent roles in the promotion of chaos

generalized inversion of finite rank hankel and toeplitz operators with rational matrix symbols abstract the goal of the paper is a generalized inversion of finite rank hankel operators and hankel or toeplitz operators with block matrices having finitely many rows to attain it a left coprime fractional factorization of a strictly proper rational matrix function and the bezout equation are used generalized inverses of these operators and generating functions for the inverses are explicitly constructed in terms of the fractional factorization

self avoiding walks in four dimensions logarithmic corrections we present simulation results for long n or 4000 self avoiding walks in four dimensions we find definite indications of logarithmic corrections but the data are poorly described by the asymptotically leading terms detailed comparisons are presented with renormalization group flow equations derived by direct renormalization and with the results of a field theoretic calculation

on some properties of positive definite toeplitz matrices and their possible applications abstract various properties of a real symmetric toeplitz matrix m with elements jk a j k 1 j k m are reviewed here matrices of this kind often arise in applications in statistics econometrics psychometrics structural engineering multichannel filtering reflection seismology etc and it is desirable to have techniques which exploit their special structure possible applications of the results related to their inverse determinant and eigenvalue problem are suggested

melnikov transforms bernoulli bundles and almost periodic perturbations in this paper we study nonlinear time varying perturbations of an autonomous vector field in the plane r2 we assume that the unperturbed equation i e the given vector field has a homoclinic orbit and we present a gen eralization of the melnikov method which allows us to show that the perturbed equation has a transversal homoclinic trajectory the key to our generalization is the concept of the melnikov transform which is a linear transformation on the space of perturbation functions the appropriate dynamical setting for studying these perturbation is the concept of a skew product flow the concept of transversality we require is best understood in this context under conditions whereby the perturbed equation admits a transversal homoclinic trajectory we also study the dynamics of the perturbed vector field in the vicinity of this trajectory in the skew product flow we show the dynamics near this trajectory can have the exotic behavior of the bernoulli shift the exact description of this dynamical phenomenon is in terms of a flow on a fiber bundle which we call the bernoulli bundle we allow all perturbations which are bounded and uniformly continuous in time thus our theory includes the classical periodic perturbations studied by melnikov quasi periodic and almost periodic perturbations as well as toroidal perturbations which are close to quasi periodic perturbations

loop erased self avoiding walks in two dimensions exact critical exponents and winding numbers abstract loop erased self avoiding walks lesaws are defined as walks resulting from sequential erasing of the loops of random walks the critical properties of lesaws are those of a c 2 conformal theory in two dimensions and geometrically those of subgraphs of spanning trees in this paper we derive the exact asymptotic behavior of the set of probabilities p k n that k k 2 lesaws of length n starting at neighbouring points do not intersect as well as the exact asymptotic winding angle distribution of a single two dimensional lesaw

generalized invertibility of hankel and toeplitz matrices abstract a result for the computation of the moore penrose inverse of hankel matrices over the field of the complex numbers is given by heinig and rost in this note some methods of characterizing and computing generalized inverses of hankel and toeplitz matrices over fields and over rings with the extended rao condition are presented the results were obtained by combining classical theory on the special structure of those classes of matrices with recent theory on generalized invertibility

a note on the representations for the drazin inverse of 2 2 block matrices abstract in 1979 campbell and meyer proposed the problem of finding a formula for the drazin inverse of a 2 2 matrix m a b c d in terms of its various blocks where the blocks a and d are required to be square matrices special cases of the problems have been studied in particular a representation of the drazin inverse of m denoted by md has recently been obtained under the assumptions that c i aad b o and a i aad b o together with the condition that the generalized schur complement d cadb be either nonsingular or zero we derive an alternative representation for md under the same assumptions but with the condition on the schur complement in the hypothesis replaced by the condition that r caa d n b n d where r and n are the range and null space of a matrix

on group inverse of singular toeplitz matrices in this paper we show that the group inverse of a real singular toeplitz matrix can be represented as the sum of products of lower and upper triangular toeplitz matrices such a matrix representation generalizes gohberg semencul formula in the literature

skew cyclic displacements and inversions of two innovative patterned matrices in this paper we deal mainly with a class of column upper plus lower cupl toeplitz matrices without toeplitz structure which are close to the toeplitz matrices in the sense that their 1 1 cyclic displacements coincide with cyclic displacement of some toeplitz matrices by constructing the corresponding displacement of the matrices we derive the formulas on representation of the inverses of the cupl toeplitz matrices in the form of sums of products of factor 1 1 circulants and 1 1 circulants furthermore through the relation between the cupl toeplitz matrices and the cupl hankel matrices the inverses of the cupl hankel matrices can be obtained as well

preconditioners for nonsymmetric block toeplitz like plus diagonal linear systems we consider the system of linear equations lu f where l is a nonsymmetric block toeplitz like plus diagonal matrix which arises from the sinc galerkin discretization of differential equations our main contribution is to construct effective preconditioners for this structured coefficient matrix and to derive tight bounds for eigenvalues of the preconditioned matrices moreover we use numerical examples to show that the new preconditioners when applied to the preconditioned gmres method are efficient for solving the system of linear equations

self organized critical state of sandpile automaton models we study a general bak tang wiesenfeld type automaton model of self organized criticality in which the toppling conditions depend on local height but not on its gradient we characterize the critical state and determine its entropy for an arbitrary finite lattice in any dimension the two point correlation function is shown to satisfy a linear equation the spectrum of relaxation times describing the approach to the critical state is also determined exactly

critical exponent for the loop erased self avoiding walk by monte carlo methods a monte carlo simulation was performed for loop erased self avoiding walks lesaw to ascertain the exponentv for the z2 and z3 lattices the estimated values were 2v 1 600 0 006 in two dimensions and 2v 1 232 0 008 in three dimensions leading to the conjecturev 4 5 for the two dimensional lesaw these results add to existing evidence that the loop erased self avoiding walks are not in the same universality class as self avoiding walks

generalized inverses of hankel and toeplitz mosaic matrices abstract hankel and toeplitz mosaic matrices are block matrices with hankel or toeplitz blocks respectively it is shown that hankel and toeplitz mosaic matrices possess reflexive generalized inverses which are bezoutians furthermore the bezoutian structure of the moore penrose and group inverses is investigated

on drazin inverse of singular toeplitz matrix in this paper we show that the drazin inverse of singular toeplitz matrix can be represented as a sum of products of lower and upper triangular toeplitz matrices and give the explicit expression which generalizes the well known result by gohberg and semencul regarding the inverse of toeplitz matrix

distribution of sizes of erased loops of loop erased random walks in two and three dimensions we show that in the loop erased random walk problem the exponent characterizing probability distribution of areas of erased loops is superuniversal in d dimensions the probability that the erased loop has an area a varies as a 2 for large a independent of d for 2 d 4 we estimate the exponents characterizing the distribution of perimeters and areas of erased loops in d 2 and 3 by large scale monte carlo simulations our estimate of the fractal dimension z in two dimensions is consistent with the known exact value 5 4 in three dimensions we get z 1 6183 0 0004 the exponent for the distribution of durations of avalanche in the three dimensional abelian sandpile model is determined from this by using scaling relations

invasion percolation a new form of percolation theory a new kind of percolation problem is described which differs from ordinary percolation theory in that it automatically finds the critical points of the system the model is motivated by the problem of one fluid displacing another from a porous medium under the action of capillary forces but in principle it may be applied to any kind of invasion process which proceeds along a path at least resistance the name invasion percolation is proposed for this new process similarities to and differences from ordinary percolation theory are discussed

structured perturbations of group inverse and singular linear system with index one in this paper we study the normwise perturbation theory of singular linear structured system with index one the structures under investigation are toeplitz circulant hankel matrices we show that the structured condition number is smaller than unstructured condition number for specific right hand side we present the condition number for structured system

scaling of waves in the bak tang wiesenfeld sandpile model n with aen ae0 n a similar ansatzes should according to this view hold for the number of topplings s which difiers from a because sites can topple more than once in an avalanche and for the radius and duration of avalanches but recent inves tigations 10 12 show that the two dimensional btw model may be characterized by a multifractal behavior where difierent moments of a are governed by exponents aen which are not linear in n and are indeed not related to each other for difierent n difierent reasons for this have been proposed in 10 11 and in 12 notice that multi fractality of avalanches can be proven for certain variants of the 1 d sandpile model 14 deviations from pure power laws had been seen already in early simulations but were usually interpreted as fl nite size efiects due to avalanches which touch the bound ary of the lattice to illustrate that this is most likely not true and that there is a real problem with simple scaling we show in fig 1 the ratio ps x l pa x l of the integrated distributions ps x l r 1 x dx 0 ps x 0 l for d 2 and for difierent values of l in these simula tions cylindrical boundary conditions were used open at y l 2 and periodic at x 0 l and data were col lected only for avalanches starting at y 0 in this way we hope to have minimized boundary efiects also since we do not make separate flts to ps x l and pa x l we have none of the uncertainties inherent in such flts due

higher correlations universal distributions and finite size scaling in the field theory of depinning recently we constructed a renormalizable field theory up to two loops for the quasi static depinning of elastic manifolds in a disordered environment here we explore further properties of the theory we show how higher correlation functions of the displacement field can be computed drastic simplifications occur unveiling much simpler diagrammatic rules than anticipated this is applied to the universal scaled width distribution the expansion in d 4 epsilon predicts that the scaled distribution coincides to the lowest orders with the one for a gaussian theory with propagator g q 1 q d 2 zeta zeta being the roughness exponent the deviations from this gaussian result are small and involve higher correlation functions which are computed here for different boundary conditions other universal quantities are defined and evaluated we perform a general analysis of the stability of the fixed point we find that the correction to scaling exponent is omega epsilon and not epsilon 3 as used in the analysis of some simulations a more detailed study of the upper critical dimension is given where the roughness of interfaces grows as a power of a logarithm instead of a pure power

scaling of loop erased walks in 2 to 4 dimensions we simulate loop erased random walks on simple hyper cubic lattices of dimensions 2 3 and 4 these simulations were mainly motivated to test recent two loop renormalization group predictions for logarithmic corrections in d 4 simulations in lower dimensions were done for completeness and in order to test the algorithm in d 2 we verify with high precision the prediction d 5 4 where the number of steps n after erasure scales with the number n of steps before erasure as n nd 2 in d 3 we again find a power law but with an exponent different from the one found in the most precise previous simulations d 1 6236 0 0004 finally we see clear deviations from the naive scaling n n in d 4 while they agree only qualitatively with the leading logarithmic corrections predicted by several authors their agreement with the two loop prediction is nearly perfect

choosing a spanning tree for the integer lattice uniformly consider the nearest neighbor graph for the integer lattice zd in d dimensions for a large finite piece of it consider choosing a spanning tree for that piece uniformly among all possible subgraphs that are spanning trees as the piece gets larger this approaches a limiting measure on the set of spanning graphs for zd this is shown to be a tree if and only if d 4 in this case the tree has only one topological end that is there are no doubly infinite paths when d 2 5 the spanning forest has infinitely many components almost surely with each component having one or two topological ends

equivalence between the abelian sandpile model and the q 0 limit of the potts model abstract we establish an equivalence between the undirected abelian sandpile model and the q 0 limit of the q state potts model the equivalence is valid for arbitrary finite graphs two dimensional abelian sandpile models thus correspond to a conformal field theory with central charge c 2 the equivalence also gives a monte carlo algorithm to generate random spanning trees we study the growth process of the spread of fire under the burning algorithm in the background of a random recurrent configuration of the abelian sandpile model the average number of sites burnt upto time t varies at ta in two dimensions our numerically determined value of a agrees with the theoretical prediction a 8 5 we relate this exponent to the conventional exponents characterizing the distributions of avalanche sizes

on preconditioned iterative methods for certain time dependent partial differential equations when the newton method or the fixed point method is employed to solve the systems of nonlinear equations arising in the sinc galerkin discretization of certain time dependent partial differential equations in each iteration step we need to solve a structured subsystem of linear equations iteratively by for example a krylov subspace method such as the preconditioned gmres in this paper based on the tensor and the toeplitz structures of the linear subsystems we construct structured preconditioners for their coefficient matrices and estimate the eigenvalue bounds of the preconditioned matrices under certain assumptions numerical examples are given to illustrate the effectiveness of the proposed preconditioning methods it has been shown that a combination of the newton fixed point iteration with the preconditioned gmres method is efficient and robust for solving the systems of nonlinear equations arising from the sinc galerkin discretization of the time dependent partial differential equations

on moore penrose inverses of toeplitz matrices abstract the moore penrose inverses of toeplitz matrices can always be represented as a sum of products of lower and upper triangular toeplitz matrices which extends the well known theorem by gohberg and semencul regarding the inverses of toeplitz matrices

the scaling limit of loop erased random walk in three dimensions we show that the scaling limit exists and is invariant under dilations and rotations we give some tools that might be useful to show universality

explicit group inverse of an innovative patterned matrix in this paper we present an innovative patterned matrix rfpl toeplitz matrix is neither the extension of toeplitz matrix nor its special case we show that the group inverse of this new patterned matrix can be represented as the sum of products of lower and upper triangular toeplitz matrices first the explicit expression of the group inverse of an rfpl toeplitz matrix is obtained second the decomposition of the group inverse is given finally an example demonstrates availability of the two methods for the group inverse

exponents and bounds for uniform spanning trees in d dimensions uniform spanning trees are a statistical model obtained by taking the set of all spanning trees on a given graph such as a portion of a cubic lattice in d dimensions with equal probability for each distinct tree some properties of such trees can be obtained in terms of the laplacian matrix on the graph by using grassmann integrals we use this to obtain exact exponents that bound those for the power law decay of the probability that k distinct branches of the tree pass close to each of two distinct points as the size of the lattice tends to infinity

a wiener filter state space flux optimal control against escape from a potential well escape from a potential well often identified with system failure is reformulated for a class of two dimensional dynamical systems as exiting the safe region of the system state space bounded by the separatrix derived from the potential well a control against system failure based on minimizing the state space flux out of this safe region is considered for systems subject to weak external forcing with available control power exceeding a certain threshold the finite lag state space flux optimal control against failure is shown to involve a wiener filter a non wiener solution is proposed for available power below this threshold these results hold for wide sense stationary forcing with any statistics and any spectrum

loop erased self avoiding random walk in two and three dimensions if n is the position of the self avoiding random walk in d obtained by erasing loops from simple random walk then it is proved that the mean square displacemente n 2 grows at least as fast as the flory predictions for the usual saw i e at least as fast asn3 2 ford 2 andn6 5 ford 3 in particular if the mean square displacement of the usual saw grows liken1 18 ind 3 as expected then the loop erased process is in a different universality class

cyclic displacements and decompositions of inverse matrices for cupl toeplitz matrices abstract in this paper we deal mainly with a class of cupl toeplitz matrices 32 without toeplitz structure which are close to the toeplitz matrices in the sense that their 1 1 cyclic displacements coincide with cyclic displacement of some toeplitz matrices by constructing the corresponding displacement of the matrices we derive the formulas on representation of the inverses of the cupl toeplitz matrices in the form of sums of products of factor row first plus last right circulants furthermore the inverses of the cupl hankel matrices can be also obtained which benefits from the relation between the cupl toeplitz matrices and the cupl hankel matrices

loop erased random walk as a spin system observable the determination of the hausdorff dimension of the scaling limit of loop erased random walk is closely related to the study of the one point function of loop erased random walk i e the probability a loop erased random walk passes through a given vertex recent work in the theoretical physics literature has investigated the hausdorff dimension of loop erased random walk in three dimensions by applying field theory techniques to study spin systems that heuristically encode the one point function of loop erased random walk inspired by this we introduce two different spin systems whose correlation functions can be rigorously shown to encode the one point function of loop erased random walk

minimum spanning trees and random resistor networks in d dimensions we consider minimum cost spanning trees both in lattice and euclidean models in d dimensions for the cost of the optimum tree in a box of size l we show that there is a correction of order l ensuremath theta where ensuremath theta ensuremath leqslant 0 is a universal d dependent exponent there is a similar form for the change in optimum cost under a change in boundary condition at nonzero temperature t there is a crossover length ensuremath xi ensuremath sim t ensuremath ensuremath nu such that on length scales larger than ensuremath xi the behavior becomes that of uniform spanning trees there is a scaling relation ensuremath theta ensuremath 1 ensuremath nu and we provide several arguments that show that ensuremath nu and ensuremath 1 ensuremath theta both equal ensuremath nu mathrm perc the correlation length exponent for ordinary percolation in the same dimension d in all dimensions d ensuremath geqslant 1 the arguments all rely on the close relation of kruskal s greedy algorithm for the minimum spanning tree percolation and for some arguments random resistor networks the scaling of the entropy and free energy at small nonzero t and hence of the number of near optimal solutions is also discussed we suggest that the steiner tree problem is in the same universality class as the minimum spanning tree in all dimensions as is the traveling salesman problem in two dimensions hence all will have the same value of ensuremath theta ensuremath 3 4 in two dimensions

phase space transport and control of escape from a potential well abstract a framework for controlling a nonlinear dynamical system against escape from a potential well is presented based on reducing the phase space transport across the separatrix associated with the potential well a bandlimited open loop control with finite lag is considered for systems with weak additive stationary forcing including specifically the colored gaussian case the related multiplicative closed loop control problem is shown to reduce to an open loop problem a numerical example based on the duffing oscillator is presented to illustrate the theory

minimum spanning trees on random networks we show that the geometry of minimum spanning trees mst on random graphs is universal because of this geometric universality we are able to characterize the energy of mst using a scaling distribution p epsilon found using uniform disorder we show that the mst energy for other disorder distributions is simply related to p epsilon we discuss the relationship to invasion percolation to the directed polymer in a random media to uniform spanning trees and also the implications for the broader issue of universality in disordered systems