random walks on combs we develop techniques to obtain rigorous bounds on the behaviour of random walks on combs using these bounds we calculate exactly the spectral dimension of random combs with infinite teeth at random positions or teeth with random but finite length we also calculate exactly the spectral dimension of some fixed non translationally invariant combs we relate the spectral dimension to the critical exponent of the mass of the two point function for random walks on random combs and compute mean displacements as a function of walk duration we prove that the mean first passage time is generally infinite for combs with anomalous spectral dimension

biased random walks on random combs we develop rigorous analytic techniques to study the behaviour of biased random walks on combs this enables us to calculate exactly the spectral dimension of random comb ensembles for any bias scenario in the teeth or spine two specific examples of random comb ensembles are discussed the random comb with nonzero probability of an infinitely long tooth at each vertex on the spine and the random comb with a power law distribution of tooth lengths we also analyse transport properties along the spine for these probability measures

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

functionals of brownian motion localization and metric graphs we review several results related to the problem of a quantum particle in a random environment in an introductory part we recall how several functionals of brownian motion arise in the study of electronic transport in weakly disordered metals weak localization two aspects of the physics of the one dimensional strong localization are reviewed some properties of the scattering by a random potential time delay distribution and a study of the spectrum of a random potential on a bounded domain the extreme value statistics of the eigenvalues then we mention several results concerning the diffusion on graphs and more generally the spectral properties of the schrodinger operator on graphs the interest of spectral determinants as generating functions characterizing the diffusion on graphs is illustrated finally we consider a two dimensional model of a charged particle coupled to the random magnetic field due to magnetic vortices we recall the connection between spectral properties of this model and winding functionals of planar brownian motion

relations between cumulants in noncommutative probability abstract we express classical free boolean and monotone cumulants in terms of each other using combinatorics of heaps pyramids tutte polynomials and permutations we completely determine the coefficients of these formulas with the exception of the formula for classical cumulants in terms of monotone cumulants whose coefficients are only partially computed

anomalous diffusion and charge relaxation on comb model exact solutions abstract the random walks on the comb structure are considered it is shown that due to fingers a diffusion has an anomalous character that is an r m s displacement depends on time by a power way with exponent 1 2 the generalized diffusion equation for an anomalous case is deduced it essentially differs from a usual diffusion equation in the continuity equation form instead of the first time derivative the time derivative of fractal order 1 2 appears in the second part the charge relaxation on the comb structure is studied a non maxwell character is established the reason is that the electric field has three components but a charge may relax only along some conducting lines

brownian excursions on combs in this paper we use techniques from ito excursion theory to analyze brownian motion on generalized combs ito excursion theory is a little known area of probability theory and we therefore present a brief introduction for the uninitiated a general method for analyzing transport along the backbone of the comb is demonstrated and the specific case of a comb whose teeth are scaling branching trees is examined we then present a recursive method for evaluating the distribution of the first passage times on hierarchical combs

1 2 order fractional fokker planck equation on comblike model from the generalized scheme of random walks on the comblike structure it is shown how a 1 2 order fractional fokker planck equation can be derived the operator method for the moments associated with the distribution function p x t is used to solve the resulting equation also the anomalous diffusion along the backbone of the structure has been considered

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

reaction diffusion wave fronts on comblike structures from the hamilton jacobi formalism an explicit expression for the speed of wave front propagation along the backbone of comblike structures is obtained this expression through the waiting time distribution function takes into account the number of sites and their distribution in the secondary branches our theoretical results are supported by numerical simulations of the reaction random walk process on the structure finally a more complex situation such as the peano basin structure is also considered both theoretically and numerically exhibiting a good agreement too

non gaussian random walks the authors present an explicit expression for the probability distribution for the position of a continuous time random walker in an arbitrary number of dimensions when the interjump density has a long time tail in contrast to earlier results which require numerical inversion of a fourier integral they replace this numerical procedure by one that relies on the method of steepest descents their results are applied to diffusion on a comb and on a percolation cluster generated on a cayley tree at criticality and are confirmed numerically

unequal twins probability distributions do not determine everything it is the common lore to assume that knowing the equation for the probability distribution function pdf of a stochastic model as a function of time tells the whole picture defining all other characteristics of the model we show that this is not the case by comparing two exactly solvable models of anomalous diffusion due to geometric constraints the comb model and the random walk on a random walk we show that though the two models have exactly the same pdfs they differ in other respects like their first passage time distributions their autocorrelation functions and their aging properties

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

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

diffusion on a random comb distribution function of the survival probability we investigate the distribution functionq p describing the survival probability on a comb consisting of a backbone with lateral randomly disconnected infinite branches two different regimes are analyzed in some detail i at short times q p is shown to have a self similar structure devil s staircase ii at large times this function becomes smooth and tends toward a rather well defined unit step function the disorder averaged survival probability is expected to decrease ast 3 4 at large times whereas the relative fluctuations of the sample dependentp0 t display a very slow decay in time going to zero liket 1 8

analysis and parameter identification of bilinear systems via shifted legendre polynomials the problems of analysis and parameter identification of lime delay systems have been recently studied using rectangular functions such as walsh and block pulse functions in this paper the continuous shifted legendre polynomials are first used to solve the above mentioned problems the approach followed is that of converting the delay differential equation to an algebraic form through using the operational matrices of integration and delay the key point is the derivation of a delay operational matrix d which relates the coefficient vector x of the shifted legendre series of a given function x t with the coefficient vector x of its delay form x t in the form x dx where the elements of a are obtained from the initial function of x t two computational examples are given to illustrate the utility of the method the availability of recursive algorithms for constructing the delay operational matrix d makes the method particularly attractive to practise

toy model of fractional transport of cancer cells due to self entrapping a simple mathematical model is proposed to study the influence of cell fission on transport the model describes fractional in time tumor development which is a one dimensional continuous time random walk the model is relevant for consideration of both solid and diffusive cancers

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

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

anomalous diffusion on a fractal mesh we present exact analytical results for properties of anomalous diffusion on a fractal mesh the fractal mesh structure is a direct product of two fractal sets one belonging to a main branch of backbones the other to the side branches of fingers both fractal sets are constructed on the entire infinite y and x axes we suggest a special algorithm in order to construct such sets out of standard cantor sets embedded in the unit interval the transport properties of the fractal mesh are studied in particular subdiffusion along the backbones is obtained with the dispersion relation x 2 t t where the transport exponent 1 has been observed as well when the environment is controlled by means of a memory kernel

some properties of a fractal time continuous time random walk in the presence of traps the ctrw has often been applied to problems related to transport in a statistically homogeneous disordered medium which means that there are no traps or reflecting boundaries to be found in the medium two physical applications one to the migration of photons in a turbid medium and the second to the theory of diffusion controlled reactions in a random medium suggest that it might be useful to study properties of the ctrw particularly as they refer to survival probability in the presence of a trap or a trapping surface we calculate a number of these properties when the pausing time density is asymptotically proportional to a stable law i e t t 1 as t t where 0 1 a forthcoming paper will establish the correspondence between properties of the ctrw and proprties of random walkers on a fractal with trapping boundaries

numerical solution of the higher order linear fredholm integro differential difference equation with variable coefficients the main aim of this paper is to apply the legendre polynomials for the solution of the linear fredholm integro differential difference equation of high order this equation is usually difficult to solve analytically our approach consists of reducing the problem to a set of linear equations by expanding the approximate solution in terms of shifted legendre polynomials with unknown coefficients the operational matrices of delay and derivative together with the tau method are then utilized to evaluate the unknown coefficients of shifted legendre polynomials illustrative examples are included to demonstrate the validity and applicability of the presented technique and a comparison is made with existing results

simulation of cavity flow by the lattice boltzmann method a detailed analysis is presented to demonstrate the capabilities of the lattice boltzmann method thorough comparisons with other numerical solutions for the two dimensional driven cavity flow show that the lattice boltzmann method gives accurate results over a wide range of reynolds numbers studies of errors and convergence rates are carried out compressibility effects are quantified for different maximum velocities and parameter ranges are found for stable simulations the paper s objective is to stimulate further work using this relatively new approach for applied engineering problems in transport phenomena utilizing parallel computers

levy processes on a generalized fractal comb comb geometry constituted of a backbone and fingers is one of the most simple paradigm of a two dimensional structure where anomalous diffusion can be realized in the framework of markov processes however the intrinsic properties of the structure can destroy this markovian transport these effects can be described by the memory and spatial kernels in particular the fractal structure of the fingers which is controlled by the spatial kernel in both the real and the fourier spaces leads to the levy processes levy flights and superdiffusion this generalization of the fractional diffusion is described by the riesz space fractional derivative in the framework of this generalized fractal comb model levy processes are considered and exact solutions for the probability distribution functions are obtained in terms of the fox h function for a variety of the memory kernels and the rate of the superdiffusive spreading is studied by calculating the fractional moments for a special form of the memory kernels we also observed a competition between long rests and long jumps finally we considered the fractal structure of the fingers controlled by a weierstrass function which leads to the power law kernel in the fourier space this is a special case when the second moment exists for superdiffusion in this competition between long rests and long jumps

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

scaling properties of diffusion on comb like structures the authors study the scaling properties of the probability density of random walks on comb like structures they find that in contrast to the first passage time in which several exponents are needed to characterise their distribution the probability density can be described by a single exponent

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

the diffusion drift equation on comb like structure from the generalized scheme of random walks on the comb like structure one can obtain the fractional fokker planck equation in the domain of fractal time evolution with critical exponent 0 1 the operator method for the moments associated with the density p x t is used to solve the obtained equation it is shown that due to fingers diffusion has an anomalous character the first two moments increase sub linearly in time

transport properties on a random comb we study the random walk of a particle in a random comb structure both in the presence of a biasing field and an the field free case we show that the mean field treatment of the quenched disorder can be exactly mapped on to a continuous time random walk ctrw on the backbone of the comb with a definite waiting time density we find an exact expression for this central quantity the green function for the ctrw is then obtained its first and second moments determine the drift and diffusion at all times we show that the drift velocity v vanishes asymptotically for power law and stretched exponential distributions of branch lengths on the comb whatever be the biasing field strength for an exponential branch length distribution v is a nonmonotonic function of the bias increasing initially to a maximum and then decreasing to zero at a critical value in the field free case anomalous diffusion occurs for a range of power law distributions of the branch length the corresponding exponent for the mean square displacement is obtained as is the asymptotic form of the positional probability distribution for the random walk we show that normal diffusion occurs whenever the mean branch length is finite and present a simple formula for the effective diffusion constant these results are extended to regular nonrandom combs as well the physical reason for anomalous drift or diffusion is traced to the properties of the distribution of a first passage time on a finite chain that controls the effective waiting time density of the ctrw

a direct approach using the shifted legendre series expansion for near optimum control of linear time varying systems with multiple state and control delays a computationally efficient method which uses finite orthogonal expansions to approximate state and control variables is developed to obtain suboptimal control for time varying systems with multiple state and control delays and with quadratic cost shifted legendre polynomials form the basis of expansions as a consequence of their useful properties the galerkin method is used to reduce the problem to one of minimizing a quadratic form with linear algebraic constraints the solution of a coupled two point boundary value problem with both delayed and advanced terms which is always required in applying the pontryagin s maximum principle to the optimization of delay systems is thus avoided

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

on the cubic velocity deviations in lattice boltzmann methods the macroscopic equations derived from the lattice boltzmann equation are not exactly the navier stokes equations here the cubic deviation terms and the methods proposed to eliminate them are studied the most popular two and three dimensional models d2q9 d3q15 d3q19 d3q27 are considered in the paper it is demonstrated that the compensation methods provide only partial elimination of the deviations for these models it is also shown that the compensation of qian and zhou 1998 europhys lett 42 359 using the compensation parameter k 1 in a d2q9 or d3q27 model can eliminate all the cross terms perfectly but the deviation terms x u3x y u3y and z u3z still survive the compensation

quantum oscillations in mesoscopic rings and anomalous diffusion we consider the weak localization correction to the conductance of a ring connected to a network we analyse the harmonics content of the al tshuler aronov spivak aas oscillations and we show that the presence of wires connected to the ring is responsible for a behaviour different from the one predicted by aas the physical origin of this behaviour is the anomalous diffusion of brownian trajectories around the ring due to the diffusion in the wires we show that this problem is related to the anomalous diffusion along the skeleton of a comb we study in detail the winding properties of brownian curves around a ring connected to an arbitrary network our analysis is based on the spectral determinant and on the introduction of an effective perimeter probing the different time scales a general expression of this length is derived for arbitrary networks more specifically we consider the case of a ring connected to wires to a square network and to a bethe lattice

lattice boltzmann method for 3 d flows with curved boundary in this work we investigate two issues that are important to computational efficiency and reliability in fluid dynamic applications of the lattice boltzmann equation lbe 1 computational stability and accuracy of different lattice boltzmann models and 2 the treatment of the boundary conditions on curved solid boundaries and their 3 d implementations three athermal 3 d lbe models q15d3 q19d3 and q27d3 are studied and compared in terms of efficiency accuracy and robustness the boundary treatment recently developed by filippova and hanel 1998 j comp phys 147 219 and mei et al 1999 j comp phys 155 307 in 2 d is extended to and implemented for 3 d the convergence stability and computational efficiency of the 3 d lbe models with the boundary treatment for curved boundaries were tested in simulations of four 3 d flows 1 fully developed flows in a square duct 2 flow in a 3 d lid driven cavity 3 fully developed flows in a circular pipe and 4 a uniform flow over a sphere we found that while the 15 velocity 3 d q15d3 model is more prone to numerical instability and the q27d3 is more computationally intensive the q19d3 model provides a balance between computational reliability and efficiency through numerical simulations we demonstrated that the boundary treatment for 3 d arbitrary curved geometry has second order accuracy and possesses satisfactory stability characteristics

numerical solution of a class of two dimensional nonlinear volterra integral equations using legendre polynomials in this paper a method for finding an approximate solution of a class of two dimensional nonlinear volterra integral equations is discussed the properties of two dimensional shifted legendre functions are presented the operational matrices of integration and product together with the collocation points are utilized to reduce the solution of the integral equation to the solution of a system of nonlinear algebraic equations some results concerning the error analysis are obtained we also consider the application of the method to the solution of certain partial differential equations illustrative examples are included to demonstrate the validity and applicability of the technique

factorization symmetry in the lattice boltzmann method abstract a non perturbative algebraic theory of the lattice boltzmann method is developed based on the symmetry of a product it involves three steps i derivation of admissible lattices in one spatial dimension through a matching condition which imposes restricted extension of higher order gaussian moments ii a special quasi equilibrium distribution function found analytically in closed form on the product lattice in two and three spatial dimensions and which proves the factorization of quasi equilibrium moments and iii an algebraic method of pruning based on a one into one relation between groups of discrete velocities and moments two routes of constructing lattice boltzmann equilibria are distinguished the present theory includes previously known limiting and special cases of lattices and enables automated derivation of lattice boltzmann models from two dimensional tables by finding the roots of one polynomial and solving a few linear systems

transfer function matrix identification in mimo systems via shifted legendre polynomials a general algorithm is presented for the identification of the parameters in the transfer function matrix of a multi input multi output mimo system the approach adopted is that of expanding the system input and output variables in shifted legendre series the feasibility of the method lies in the generation of linear algebraic equations in the unknown parameters and initial conditions by means of an elegant operational matrix which relates shifted legendre polynomials to their integrals an example is included to illustrate the applicability of the proposed method

speeding up evolutionary search by small fitness fluctuations we consider a fixed size population that undergoes an evolutionary adaptation in the weak mutation rate limit which we model as a biased langevin process in the genotype space we show analytically and numerically that if the fitness landscape has a small highly epistatic rough and time varying component then the population genotype exhibits a high effective diffusion in the genotype space and is able to escape local fitness minima with a large probability we argue that our principal finding that even very small time dependent fluctuations of fitness can substantially speed up evolution is valid for a wide class of models

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

analytical solutions for diffusion on fractal objects a generalization of the euclidean diffusion equation is proposed for diffusion on fractals on the basis of a scaling argument a renormalization group theory for the green s function and numerical tests conjectures on the applicability to natural fractals such as porous media are presented

transport and fluctuation dissipation relations in asymptotic and preasymptotic diffusion across channels with variable section we study the asymptotic and preasymptotic diffusive properties of brownian particles in channels whose section varies periodically in space the effective diffusion coefficient d mathrm eff is numerically determined by the asymptotic behavior of the root mean square displacement in different geometries considering even cases of steep variations of the channel boundaries moreover we compared the numerical results to the predictions from the various corrections proposed in the literature to the well known fick jacobs approximation building an effective one dimensional equation for the longitudinal diffusion we obtain an approximation for the effective diffusion coefficient such a result goes beyond a perturbation approach and it is in good agreement with the actual values obtained by the numerical simulations we discuss also the preasymptotic diffusion which is observed up to a crossover time whose value in the presence of strong spatial variation of the channel cross section can be very large in addition we show how the einstein s relation between the mean drift induced by a small external field and the mean square displacement of the unperturbed system is valid in both asymptotic and preasymptotic regimes