some properties of a random walk on a comb structure abstract we analyze transport properties of a random walk on a comb structure which serves as a model for a random walk on the backbone of a percolation cluster it is shown that the random walk along the x axis which is the analog of the backbone exhibits anomalous diffusion in that x 2 n n 1 2 and the expected number of x sites visited is proportional to n 1 4 for large n the distribution function is found to be a two dimensional gaussian if a field in the x direction so that diffusion is asymmetric the expected displacement is found to be asymptotically proportional to n 1 2

equivalence and normal forms of bilinear forms abstract we present an alternative account of the problem of classifying and finding normal forms for arbitrary bilinear forms beginning from basic results developed by riehm our solution to this problem hinges on the classification of indecomposable forms and in how uniquely they fit together to produce all other forms we emphasize the use of split forms i e those bilinear forms such that the minimal polynomial of the asymmetry of their non degenerate part splits over ground field rather than restricting the field to be algebraically closed in order to obtain the most explicit results without resorting to the classification of hermitian symmetric and quadratic forms we merely require that the underlying field be quadratically closed

on normal subgroups of multiplicative group of skew fields generated by a polycyclic by finite group 1 1 let d be a field and let d be its multiplicative group consider a subgroup g c d and a subring k of the center of d we denote the subfield the term field in this paper is used in the skew field sense generated by k and g by k g when k is the subring generated by the unit elements i e k z or k z we simply say that d is generated by g the main result of the article is the following theorem

unified continuum description for sub diffusion random walks on multi dimensional comb model abstract the anisotropic sub diffusion random walks on multi dimensional comb structure model have been studied in the continuum approximation the problem is that in the considered model mean square displacements on different directions have different power temporal dependencies so this case essentially differs from usual anisotropic diffusion case and it is not obviously how to describe such a diffusion by a unified way nevertheless the unified continuum description for such random walks has been developed as result the new generalization of fick s law for diffusion current has been obtained for anisotropic anomalous sub diffusion random walks in multi dimensional case namely it has been shown that the effective diffusion coefficient in the generalized fick s law has new operator form components of diffusion tensor instead of constant values have the fractional order temporal derivatives

congruences of a square matrix and its transpose abstract it is known that any square matrix a over any field is congruent to its transpose at stas for some nonsingular s moreover s can be chosen such that s2 i that is s can be chosen to be involutory we show that a and at are congruent over any field f of characteristic not two with involution a a the involution can be the identity a t s t as for some nonsingular s moreover s can be chosen such that s s i that is s can be chosen to be coninvolutory the short and simple proof is based on sergeichuk s canonical form for congruence math ussr izvestiya 31 3 1988 481 it follows that any matrix a over f can be represented as a eb in which e is coninvolutory and b is symmetric

first encounters on combs we consider two random walkers embedded in a finite two dimension comb and we study the mean first encounter time mfet evidencing mainly numerically different scalings with the linear size of the underlying network according to the initial position of the walkers if one of the two players is not allowed to move then the first encounter problem can be recast into a first passage problem mfpt for which we also obtain exact results for different initial configurations by comparing mfet and mfpt we are able to figure out possible search strategies and in particular we show that letting one player be fixed can be convenient to speed up the search as long as we can finely control the initial setting while for a random setting on average letting one player rest would slow down the search

the full martin boundary of the bi tree are interested in the martin compactification of the bi tree with respect to each eigenvalue t p of the transition operator acting on nonnegative functions the minimal martin boundary i e set of minimal t harmonic functions is well understood see 20 and also 22 for cartesian products of markov chains in general on the other hand the description of the full martin compactification has remained an open problem some evidence has been obtained by picardello and sjogren 21 who studied boundary behaviour of harmonic functions on the bi tree the situation is similar for the hyperbolic bi disk equipped with the laplace beltrami operator in this case the minimal martin boundary is known via the work of karpelevic 13 and guivarc h 10 see also the general result of freire 7 and taylor 26 concerning cartesian products of manifolds the problem of determining the full martin boundary was proposed by guivarc h and taylor 11 who gave the answer only for the critical eigenvalue where the behaviour is different in the present paper we describe the full martin compactification of the bitree with respect to all eigenvalues t p this is possible thanks to recent work of lalley 14 who has given a local limit theorem for aperiodic nearestneighbour random walks on homogeneous trees that is an asymptotic estimate of the n step transition probabilities which is uniform both in time and space a considerable improvement of previous work of sawyer 24 gerl and woess

generalized diffusion equation for anisotropic anomalous diffusion motivated by studies of comblike structures we present a generalization of the classical diffusion equation to model anisotropic anomalous diffusion we assume that the diffusive flux is given by a diffusion tensor acting on the gradient of the probability density where each component of the diffusion tensor can have its own scaling law we also assume scaling laws that have an explicit power law dependence on space and time solutions of the proposed generalized diffusion equation are consistent with previously derived asymptotic results for the probability density on comblike structures

diffusion on random comb structure effective medium approximation abstract the random walks on comb model of percolation clusters are studied the generalized diffusion equations in the form of fractal temporal derivatives are deduced the random distributions of fingers over lengths gaussian and power are studied in the effective medium approximation

occupation times on a comb with ramified teeth we investigate occupation time statistics for random walks on a comb with ramified teeth this is achieved through the relation between the occupation time and the first passage times statistics of occupation times in half space follows lamperti s distribution i e the generalized arcsine law holds transitions between different behaviors are observed which are controlled by the size of the backbone and teeth of the comb as well as bias occupation time on a nonsimply connected domain is analyzed with a mean field theory and numerical simulations in that case the generalized arcsine law is not valid

free algebras and free groups in ore extensions and free group algebras in division rings abstract let k be a field of characteristic zero let be an automorphism of k and let be a derivation of k we show that the division ring d k x either has the property that every finitely generated subring satisfies a polynomial identity or d contains a free algebra on two generators over its center in the case when k is finitely generated over a subfield k we then see that for a k algebra automorphism of k and a k linear derivation of k k x having a free subalgebra on two generators is equivalent to having infinite order and k x having a free subalgebra is equivalent to being nonzero as an application we show that if d is a division ring with center k of characteristic zero and d contains a solvable subgroup that is not locally abelian by finite then d contains a free k algebra on two generators moreover if we assume that k is uncountable without any restrictions on the characteristic of k then d contains the k group algebra of the free group of rank two

mesoscopic description of random walks on combs combs are a simple caricature of various types of natural branched structures which belong to the category of loopless graphs and consist of a backbone and branches we study continuous time random walks on combs and present a generic method to obtain their transport properties the random walk along the branches may be biased and we account for the effect of the branches by renormalizing the waiting time probability distribution function for the motion along the backbone we analyze the overall diffusion properties along the backbone and find normal diffusion anomalous diffusion and stochastic localization diffusion failure respectively depending on the characteristics of the continuous time random walk along the branches and compare our analytical results with stochastic simulations

scaling of the first passage time and the survival probability on exact and quasi exact self similar structures the authors calculate the asymptotic behaviour of the moments of the first passage time and survival probability for random walks on an exactly self similar tree and on a quasi self similar comb by applying an exact decimation approach to the master equations for the hierarchical comb a transition from ordinary to anomalous diffusion occurs at r 2 where r is the ratio of teeth length in successive iterations of the structure in the anomalous regime r 2 the positive integer moments of the first passage time tq scale as ltau q with tau q 1 2q 1 ln r ln 2 where l is the linear distance from input to output the asymptotic behaviour of the survival probability is studied using both scaling theory and by a direct solution of the master equations they find that the characteristic time t in the asymptotic exponential decay of the survival probability exp t t scales as t approximately ltau with tau ln r2 ln 2 i e tau is distinct from tau 1 however substantial corrections to this asymptotic form for tau exist and these are needed to account for the recent simulation data of havlin and matan 1988

exact renormalisation group approach to ultradiffusion in a hierarchical structure a model of diffusion in one dimension with a hierarchical pattern of hopping rates is studied by an exact renormalisation method non universal time scaling exponents are obtained for the autocorrelation function range and average diffusion distance

congruence and conjunctivity of matrices to their adjoints abstract every square matrix over a field f is involutorily congruent over f to its transpose and hence each such matrix is the product of a symmetric matrix and an involutory matrix over f in the usual complex case every matrix which is conjunctive with its adjoint conjugate transpose is involutorily conjunctive with its adjoint and hence is the product of a hermitian matrix and an involutory matrix furthermore every such matrix is conjunctive with a real matrix these three conditions on a matrix 1 being conjunctive with its adjoint 2 being involutorily conjunctive with its adjoint and 3 being conjunctive with a real matrix are studied in the more general context of a field f with involution and it is shown in general that 3 implies 2 that 2 implies 3 if char f 2 a 2 2 counterexample exists for each f with char f 2 and that 1 does not in general imply 2 a 2 2 counterexample in the complexification of the rational field is presented the problem of deciding which matrices satisfy 2 is equivalent even in this general context to the problem of deciding which pairs of self adjoint hermitian matrices are involutorily conjunctive for the general 2 2 case the three conditions are characterized in terms of norms

generalized time dependent schrodinger equation in two dimensions under constraints we investigate a generalized two dimensional time dependent schrodinger equation on a comb with a memory kernel a dirac delta term is introduced in the schrodinger equation so that the quantum motion along the x direction is constrained at y 0 the wave function is analyzed by using green s function approach for several forms of the memory kernel which are of particular interest closed form solutions for the cases of dirac delta and power law memory kernels in terms of fox h function as well as for a distributed order memory kernel are obtained further a nonlocal term is also introduced and investigated analytically it is shown that the solution for such a case can be represented in terms of infinite series in fox h functions green s functions for each of the considered cases are analyzed and plotted for the most representative ones anomalous diffusion signatures are evident from the presence of the power law tails the normalized green s functions obtained in this work are of broader interest

field induced drift and trapping in percolation networks the authors study the effects of a bias producing external field on a random walk on the infinite cluster in the percolation problem there are two competing physical effects drift and trapping which result in a drift velocity nu which rises and then falls as the field increases they study these effects on a one dimensional lattice with random length branches and on the diluted bethe lattice they calculate nu in the first model with a maximum allowed length for each branch in the limit that this cutoff length becomes infinite they find that the velocity vanishes identically above a finite critical value of the field for the bethe lattice they derive an upper bound on the critical field which varies as p pc 1 2 as the percolation concentration pc is approached in the one dimensional model they also investigate the anomalous regime in which the velocity vanishes they discuss the distribution of steady state times required to traverse n sites and find that it can be described in terms of a stable distribution of index x with superimposed oscillations the index x of the stable distribution is given by l xi where xi is a characteristic branch length and l is a bias induced length which describes the exponential buildup of the steady state density of particles towards the end of a branch

random walks on braid groups brownian bridges complexity and statistics we investigate the limit behaviour of random walks on some non commutative discrete groups related to knot theory namely we study the connection between the limit behaviour of the lyapunov exponent of products of non commutative random matrices generators of the braid group and the asymptotics of powers of the algebraic invariants of randomly generated knots we turn the simplest problems of knot statistics into the context of random walks on hyperbolic groups we also consider the limit distribution of brownian bridges on so called locally non commutative groups

global first passage times of fractal lattices the global first passage time density of a network is the probability that a random walker released at a random site arrives at an absorbing trap at time t we find simple expressions for the mean global first passage time for five fractals the d dimensional sierpinski gasket t fractal hierarchical percolation model mandelbrot given curve and a deterministic tree we also find an exact expression for the second moment and show that the variance of the first passage time var t scales with the number of nodes within the fractal n such that var t similar to n 4 d where d is the spectral dimension

free products of units in algebras i quaternion algebras abstract letabe a quaternion algebra over a commutative unital ring we find sufficient conditions for pairs of units ofato generate a free group using the well known isomorphism betweenso 3 r and the group of real quaternions of norm 1 we obtain free groups of rotations of the euclidean 3 space specialization techniques allow us to find similar free subgroups in skew polynomial rings a consequence is the following letkgbe the group algebra of a residually torsion free nilpotent groupgover a fieldkwhose characteristic is not 2 ifxandyare any pair of noncommuting elements ofg andc d k then 1 cxand 1 dygenerate a free subgroup of the malcev neumann field of fractions ofkg

propagators of random walks on comb lattices of arbitrary dimension we study diffusion on comb lattices of arbitrary dimension relying on the loopless structure of these lattices and using first passage properties we obtain exact and explicit formulae for the laplace transforms of the propagators associated to nearest neighbour random walks in both cases where either the first or the last point of the random walk is on the backbone of the lattice and where the two extremities are arbitrarily chosen as an application we compute the mean square displacement of a random walker on a comb of arbitrary dimension we also propose an alternative and consistent approach of the problem using a master equation description and obtain simple and generic expressions of the propagators this method is more general and is extended to study the propagators of random walks on more complex comb like structures in particular we study the case of a two dimensional comb lattice with teeth of finite length

diffusion on random comblike structures field induced trapping effects an analytic study of diffusion in random comblike structures in which field induced trapping effects enter into play is presented in order to mimic at best a percolation cluster above pc the distribution of the lengths of the branches is taken as exponential the branches act as traps of efficiency growing with the applied bias the asymptotic behaviour at large time of the probability of presence of the particle at its initial site and on the backbone and of its position and dispersion are computed in an exact manner as a result when the field exceeds a certain critical value anomalous behaviours emerge characterized by exponents depending continuously on the field

matrix rings and linear groups over a field of fractions of enveloping algebras and group rings i 1 1 let h be a lie algebra over a field k u h be its universal envelope it is well known that u h is a domain see 1 section v 31 and the theorem of cohn see 2 states that u h can be imbedded in a field when h is locally finite u h has even a field of fractions see 1 theorem v 3 61 the class of lie algebras whose universal envelope has a field of fractions is however much wider than the class of locally finite lie algebras it follows from proposition 4 1 of the article that it contains for instance all the locally soluble by locally finite algebras this gives a negative answer of the question in 2 page 528 let h satisfy any one of the following three conditions

free groups in normal subgroups of the multiplicative group of a division ring abstract let d be a division ring with center z and multiplicative group d 0 d and let n be a normal subgroup of d we investigate various conditions under which n must contain a free noncyclic subgroup in one instance assuming that the transcendence degree of z over its prime field is infinite and that n contains a nonabelian solvable subgroup we use a construction method due to chiba to exhibit free generators of the free subgroup

free products of units in algebras ii crossed products abstract we investigate the problem of explicitly constructing non cyclic free groups in finite dimensional crossed products using valuation criteria the results are applied to produce explicit free groups in division algebras generated by nilpotent groups and symmetric free groups in group rings of finite groups in arbitrary characteristic

continuum diffusion on networks trees with hyperbranched trunks and fractal branches the probability that a random walker returns to its origin for large times scales as t d over 2 where d over is the spectral dimension we calculate d over for a class of tree structures using a renormalization technique on an infinite continued fraction we consider a wide range of homogeneous networks based on replacing the branches of a self similar tree with arbitrary fractals and composite fractals we also consider a new class of inhomogeneous hyperbranched trees

equidistribution of random walks on spheres we study the equidistribution on spheres of the n step transition probabilities of random walks on graphs we give sufficient conditions for this property being satisfied and for the weaker property of asymptotical equidistribution we analyze the asymptotical behaviour of the green function of the simple random walk on 2 and we provide a class of random walks on cayley graphs of groups whose transition probabilities are not even asymptotically equidistributed

validity of the mean field approximation for diffusion on a random comb we consider unbiased diffusion on a random comb structure an infinitely long backbone with loopless branches of arbitrary length emanating from it if t ifmmode pm else textpm fi j 0 mathrm mathit w mathit t 0 is the mean time averaged over all random walks for first passage from an arbitrary origin 0 on the backbone to either of the sites j or j on it in a given realization of the structure the exact diffusion constant for the problem is defined as k mathrm lim mathit j ensuremath rightarrow mathrm ensuremath infty mathit j 2 1 mathit t 0 mathrm mathit c where mathrm mathit c stands for the configuration average over the realizations of the random comb the diffusion constant in the mean field approximation is given by mathit k mathrm mf mathrm lim mathit j ensuremath rightarrow mathrm ensuremath infty mathit j 2 mathit t 0 mathrm mathit c we compute mathit t 0 exactly for an arbitrary realization of the comb and then show rigorously that owing to the suppression of the relative fluctuations in mathit t 0 in the thermodynamic limit j ensuremath rightarrow ensuremath infty we have mathit k mathrm mf k whenever the moments of certain random variables ensuremath gamma l ensuremath alpha ensuremath beta are finite here the site dependent random variables l ensuremath alpha and ensuremath beta are respectively the branch length stay probability at the tip of a branch and the backbone to branch jump probability finally we discuss different situations in which k will not be equal to mathit k mathrm mf although the transport remains diffusive as opposed to those in which anomalous diffusion occurs textcopyright 1996 the american physical society