random walk on fractals numerical studies in two dimensions monte carlo calculations are used to investigate some statistical properties of random walks on fractal structures two kinds of lattices are used the sierpinski gasket and the infinite percolation cluster in two dimensions among other problems the authors study i the range rn of the walker number of distinct visited sites during n steps average value sn variance sigma n and asymptotic distribution ii renewal theory return to the original site probability of return p0 n mean number of returns nu n the probability distribution of the walker position p n r after n steps is discussed the asymptotic behaviour n 1 of these quantities exhibits power laws with associated exponents the numerical values of these exponents are in good agreement with recent theoretical predictions alexander and orbach 1982 rammal and toulouse 1982

multiresolution wavelet coarsening and analysis of transport in heterogeneous media abstract modeling transport in a disordered medium with broadly distributed and correlated heterogeneities has been hampered by the fact that a very large set of equations must be solved if the heterogeneities are distributed over several distinct length scales we recently proposed a novel method based on the application of wavelet transformations phys rev lett 79 1997 4385 that can be used for simulation of transport processes in such disordered media in this paper we extend the method so as to be able to study transport in two classes of highly heterogeneous media that are otherwise very difficult to simulate namely disordered anisotropic media and those that contain more than one distinct family of transport paths for example natural rock with both pores and fractures we report the results of extensive computer simulations of transport in such disordered media in which the heterogeneities are broadly distributed and contain long range correlations we show that even for the type of disordered media that we consider in this paper the wavelet based method drastically reduces by at least two orders of magnitude the number of equations to be solved without neglecting any important information about the media s morphology hence providing highly accurate estimates of their transport properties thus we conclude that the wavelet based method is applicable to any disordered medium with any type of heterogeneities

algebraic and scattering aspects of a symmetric solvable potential we study a particular solvable potential and analyse the effect of symmetry on its bound state as well as scattering solutions we determine the transmission and reflection coefficients for the symmetric case and also formulate the problem in terms of an su 1 1 potential group which allows unified treatment of the discrete and the continuous spectra in a natural way we find that bound and scattering states of the symmetric problem supply a basis for the unitary irreducible representations of the su 1 1 potential group and this gives a straightforward group theoretical interpretation of the fact that the complex invariant potential has a real energy spectrum

a new method for the calculation of the conductivity of inhomogeneous systems a new method for computing the conductivity of random irregular resistor networks is developed this method is a generalization of the transfer matrix technique proposed by derrida and vannimenus for regular 2d and 3d lattices at the same time for large systems the method presented in this paper is more efficient than the transfer matrix technique to demonstrate the method it is applied to a cubic lattice at the percolation threshold and away from it the conductivity has been found for lattices with size up to 324 3 the ratio between the conductivity exponent t and the correlation length exponent was estimated to be t 2 315 in good agreement with the literature data

percolation conductivity of penrose tiling by the transfer matrix monte carlo method abstract a generalization of the derrida and vannimenus transfer matrix monte carlo method has been applied to calculations of the percolation conductivity in a penrose tiling strips with a length 104 and widths from 3 to 19 have been used disregarding the differences for smaller strip widths up to 7 the results show that the percolative conductivity of a penrose tiling has a value very close to that of a square lattice the estimate for the percolation transport exponent once more confirms the universality conjecture for the 0 1 distribution of resistors

study of discontinuities in eigenvalue spectra of some model hamiltonians in this paper the authors study the eigenvalue spectra of two model s wave hamiltonians in three dimensions h p2 2 1 r 2 mu r 2 lambda 2r2 and h p2 2 2 mu r 2 lambda 2r2 using the method of hill determinants they show that these eigenvalue spectra display discontinuities in the limit when both coupling constants mu and lambda vanish simultaneously they use a simple variational calculation to argue that such discontinuities are characteristic of these hamiltonians and are not an artefact of their numerical methods in fact such discontinuities are also to be seen in the case of a displaced harmonic oscillator in one dimension whose eigenvalue spectrum is exactly solvable

the remarkable properties of the associated romanovski functions for and with n m q 0 a non negative integer we introduce a differential equation for the associated romanovski functions r q n m x so that its form remains invariant under the transformation n 2q n 1 introducing the rodrigues formula for both regions and we obtain simultaneously and separately realization of the laddering equations with respect to the labels n and m by means of two pairs of ladder operators also the square integrability and orthogonality of the functions are considered in cases 0 and 0 with respect to the weight functions and 1 x2 1 q respectively exponential generating functions for the infinite sequences with the same n in both regions and as well as for the infinite ones with given even and odd values of n m q are separately calculated as an application to supersymmetric quantum mechanics we show that there exist four different methods for generating the spectrum of the noncentral hyperbolic scarf potential

elastic properties of diffusion limited aggregates a transfer matrix method has been used to investigate the power law dependence of the effective elastic moduli of a two dimensional diffusion limited aggregate on the size l of the aggregate scaling exponents have been found to depend on the type of applied deformation for stretching along the direction of growth the elastic modulus scales as l 2 2 or 0 2 while for deformations in the direction perpendicular to the direction of growth it is proportional to l 2 8 or 0 2

self avoiding walks which cross a square the authors consider self avoiding walks on the square lattice which are confined to lie in or on the boundary of a square with vertices at 0 0 0 l l 0 and l l they ask for the number of such walks which begin at the origin and end at the vertex l l especially in the large l limit similarly they ask for the mean number of steps in such walks as a function of l at fixed l the authors also associate a fugacity with the number of steps of the walk and ask how the system behaves as a function of this fugacity they provide some rigorous results in particular proving that there is a phase transition at some particular value of the fugacity and supplement these with the analysis of series data for the problem

reflectionless potentials for difference schrodinger equations as a part of the program discrete quantum mechanics we present general reflectionless potentials for difference schrodinger equations with pure imaginary shifts by combining contiguous integer wave number reflectionless potentials we construct the discrete analogues of the potential with the integer h which belong to the recently constructed families of solvable dynamics having the q ultraspherical polynomials with as the main part of the eigenfunctions for the general scattering theory for these potentials we need the connection formulas for the basic hypergeometric function with which is not known the connection formulas are expected to contain the quantum dilogarithm functions as the counterparts of the q gamma functions we propose a conjecture of the connection formula of the function with based on the conjecture we derive the transmission and reflection amplitudes which have all the desirable properties they provide a strong support to the conjectured connection formula

on the organisation of transient chaos application to irregular scattering it is shown how the local structure of chaotic repellers being responsible for transient chaotic behaviour is deduced from the properties of hyperbolic periodic orbits relations between static and dynamical multifractal spectra with respect to the natural invariant measure on the repeller are derived for invertible maps of the plane the results obtained for maps with unit jacobian apply to hamiltonian systems with two degrees of freedom which exhibit the phenomenon of irregular scattering and are characterised by an exponential decay of trapping probability

robust algorithm for random resistor networks using hierarchical domain structure in this study we discuss methods for solving random resistor networks and similar problems we discuss the node elimination method and we demonstrate its equivalence to the gaussian elimination scheme finding a good elimination order which makes the method highly efficient the transfer matrix method is shown to be a special case of the node elimination method with an ordering that is far from optimal we compare the performance of these exact methods with a state of the art conjugate gradient solver in general the node elimination method is the faster method

diffusion in random media as a problem of interacting bose and fermi fields diffusion on a percolative lattice is formulated as a theory of interacting bose fermi fields the effective medium approximation ema of odagaki and lax 1981 is obtained directly from the functional integral and is used to provide an ema approximation for the diffusion constant and the dc conductivity simple corrections to the usual ema produce exponents near the percolation threshold that agree with the best available data for d 2 and 3 this agreement reveals a strong connection between the dynamics of percolative diffusion and percolation itself

recursion transform method for computing resistance of the complex resistor network with three arbitrary boundaries we develop a general recursion transform r t method for a two dimensional resistor network with a zero resistor boundary as applications of the r t method we consider a significant example to illuminate the usefulness for calculating resistance of a rectangular m ifmmode times else texttimes fi n resistor network with a null resistor and three arbitrary boundaries a problem never solved before since green s function techniques and laplacian matrix approaches are invalid in this case looking for the exact calculation of the resistance of a binary resistor network is important but difficult in the case of an arbitrary boundary since the boundary is like a wall or trap which affects the behavior of finite network in this paper we obtain several general formulas of resistance between any two nodes in a nonregular m ifmmode times else texttimes fi n resistor network in both finite and infinite cases in particular 12 special cases are given by reducing one of the general formulas to understand its applications and meanings and an integral identity is found when we compare the equivalent resistance of two different structures of the same problem in a resistor network

note on the darboux transformation for the derivative nonlinear schrodinger equation the darboux and backlund transformations for the derivative nonlinear schrodinger equation are derived by the same method for the nonlinear schrodinger equation

a solvable model of quantum discontinuities it is shown that an appropriate triple well version of the binding potential v x ax2 bx4 cx6 a c 0 b 0 b2 3ac provides a solvable model whereby the discontinuous behaviour of the energy eigenvalues of any finite set of lowest lying levels can be deduced analytically the reason for the occurrence of such discontinuities is discussed

cumulant renormalisation group and its application to the incipient infinite cluster in percolation the authors introduce the cumulant method into the context of position space renormalisation group the advantage of this method over previous procedures is that one need not systematically evaluate all 2n states of each kadanoff cell where n is the number of elements sites or bonds in the cell they illustrate this method by calculating for bond percolation with an eight bond cell the critical exponents characterizing the following quantities i the mean number of backbone bonds ii the mean number of bonds in the minimum and maximum paths along the backbone iii the mean number of singly connected red bonds and iv the mean number of bonds in all the self avoiding walks connecting the two extreme points they derive previously obtained results of the conductivity and resistivity

superuniversal spectral dimension for dilute branched polymers the authors introduce the problem of random walks on lattice animals dilute branched polymers and calculate the spectral dimension ds of animals in two and three dimensions by using a two parameter position space renormalisation group method this problem which is a generalisation of the problem of the ant in the labyrinth has been independently proposed and investigated by wilke et al 1984 the spectral dimension is given by ds 2da dw where da is the fractal dimension of the lattice animal and dw the fractal dimension of the random walk on the lattice animal the results indicate that ds may be a superuniversal quantity i e its value is independent of dimension moreover the authors find ds to be close to 4 3 which is the spectral dimension of the largest percolation cluster at the percolation threshold pc as conjectured by alexander and orbach 1982 since both lattice animals and the largest percolation cluster at pc have a homogeneous interior structure the results suggest that the spectral dimension of all such fractals may equal 4 3

learning mutation and long run equilibria in games an evolutionary model with a finite number of players and with stochastic mutations is analyzed the expansion and contraction of strategies is linked to their current relative success but mutuation perturbing the system from its deterministic evolution are present as well the focus is on the long run implications of ongoing mutations which drastically reduce the set of equilibria for 2 by 2 symmetric games with two symmetric strict nash equilibria the risk dominant equilibrium is selected in particular if both strategies have equal security levels the pareto dominant nash equilibrium is selected in particular if both strategies have equal security levels the pareto dominant nash equilibrium is selected even though there is another strict nash equilibrium copyright 1993 by the econometric society

isospectral problem interplay between liouville equations darboux transforms and mckean trubowitz flows systems of liouville equations are generated by isospectral transforms of the schrodinger operator and describe the isospectral evolution of both a potential and eigenfunctions relations of these equations with the mckean trubowitz isospectral flows are studied the role of these flows among other isospectral flows is discussed

the hierarchy of multi soliton solutions of the derivative nonlinear schrodinger equation we provide a relatively simple approach to backlund transformations for the derivative nonlinear schrodinger equation by iteration it leads to compact n soliton formulae both with asymptotically vanishing and non vanishing amplitudes the phenomenology of these solutions is discussed and illustrated in some detail

ac response near percolation threshold transfer matrix calculation in 2d the complex admittivity of a network of resistors and capacitors is calculated at pc on a square lattice using a transfer matrix the loss tangent shows a region of anomalous frequency scaling with t s finite size scaling reveals two crossover frequencies as expected from duality

effects of pair correlation on mean field theory of btw sand pile model we extend the mean field calculation of btw sand pile model to one that includes the correlation between pairs of nearest neighbors specifically we derive dynamical equations of both one site and two site densities and solve the equations order by order starting with the mean field solution the investigation provides analytical results for both stationary and dynamic states of the sand pile near the critical point which are valid in the regime where h e2 1h e2 1 h incoming rate of sand grains e bulk dissipation rate of sand grainse bulk dissipation rate of sand grains in the stationary case we evaluate the pair correlation and the correction to the mean field single site densities due to the correlation the correction is found to be of the same order as the mean field solution in the dynamic case the initial state deviates from the stationary state by a small fluctuation which subsequently decays exponentially with the time constant being reduced from the corresponding mean field value again the correction to the time constant in this case is found comparable to the mean field value itself

some exact results for moments of 2d directed animals using computer enumerations and the algebraic approximant method of series analysis several new exact results have been found for moments of width perimeter and loops for directed animals on the square and triangular lattices a proof using q series is given for exact solutions for widths on both the square and triangular lattices

anisotropic bond percolation by position space renormalisation group we treat a percolation model with anisotropic bond occupation probabilities in order to study the crossover behaviour of the effective spatial dimension of the system previous position space renormalisation group psrg studies of this problem on the square lattice show a crossover from two dimensional to pseudo one dimensional critical behaviour to investigate the possible reasons for this surprising result which is the opposite of the behaviour observed in thermal critical phenomena we first develop alternative psrg schemes we study the predictions of these groups for asymptotically large rescaling parameters where the approximations in the psrg are thought to become negligible from a consideration of the approximations involved we are led to a decimation transformation that uses an anisotropic cluster the results of this method indicate that anisotropic percolation is in the same universality class as isotropic percolation in complete analogy with thermal critical phenoma our results indicate several peculiar features both of this problem and of the psrg methods

precise calculation of the dynamical exponent of two dimensional percolation we report numerical data obtained on the special purpose computer percola for the exponent tilde t of the electrical conductivity of 2d percolation the extrapolation yields tilde t 0 9745 pm 0 0015 and a correction to the scaling exponent 1 2 0 2

higher order susy linearized nonlinear heisenberg algebras and coherent states using an iterative construction of the first order intertwining technique we find k parametric families of exactly solvable anharmonic oscillators whose spectra consist of a part isospectral to the oscillator plus k additional levels at arbitrary positions below e0 it is seen that the natural ladder operators for these systems give place to polynomial nonlinear algebras and it is shown that these algebras can be linearized the coherent states construction is performed in the nonlinear and linearized cases

unbinding transition in a many string system a 1 1 dimensional model of n nonintersecting strings with short range attractive interactions on a lattice is solved exactly for arbitrary finite n 2 there is a second order binding unbinding transition with the same critical exponents as for n 2 in the limit n to infinity the transition becomes first order

a mixed transfer matrix method for simulating normal conductor perfect insulator perfect conductor random networks we develop a mixed transfer matrix approach for computing the macroscopic conductivity of a three constituent normal conductor perfect insulator perfect conductor random network this is applied to two dimensional and three dimensional samples at a percolation threshold such networks are simulated in order to test whether a diluted percolating network of normal conducting bonds remains in the same universality class of critical behavior when a finite fraction of those bonds have been replaced by perfectly conducting bonds also tested by such simulations is whether a percolating mixture of normal and perfectly conducting bonds remains in the same universality class of critical behavior when a finite fraction of the normal bonds are replaced by perfectly insulating bonds these questions are crucial for some recently published exact results which connect the macroscopic electrical and elastic responses of percolating networks

asymptotic expansion for the resistance between two maximally separated nodes on an m by n resistor network we analyze the exact formulae for the resistance between two arbitrary notes in a rectangular network of resistors under free periodic and cylindrical boundary conditions obtained by wu j phys a 37 6653 2004 based on such expression we then apply the algorithm of ivashkevich izmailian and hu j phys a 35 5543 2002 to derive the exact asymptotic expansions of the

a transfer matrix study of conductivity and permeability exponents in continuum percolation the conductivity and permeability exponents for the swiss cheese model of continuum percolation are calculated numerically using the transfer matrix method in two dimensions the conductivity exponent t is shown to equal its universal value tun 1 24 while the permeability exponent e 2 53 is considerably larger the same exponents in three dimensions are also determined and found to be t 2 46 and e 4 1 greatly exceeding the universal value tun 1 95 the results are in fair agreement with earlier theoretical predictions

duality mapping and unbinding transitions of semiflexible and directed polymers directed polymers strings and semiflexible polymers filaments are one dimensional objects governed by tension and bending energy respectively they undergo unbinding transitions in the presence of a short range attractive potential using transfer matrix methods we establish a duality mapping for filaments and strings between the restricted partition sums in the absence and the presence of a short range attraction this allows us to obtain exact results for the critical exponents related to the unbinding transition the transition point and transition order

tricritical points of trails their euler digraphs and graphs exact results on the sierpinski gasket the collapse transitions of trails their oriented graphs and silhouettes eulerian digraphs and graphs respectively as the fugacity for crossings is increased are investigated by exact decimation on the 2d sierpinski gasket recursion relations between the generating functions for the three basic configurations on consecutive levels are derived for all models the authors find tricritical points which move along a line in a four dimensional parameter space as the fugacity is varied and terminate at a decoupled first or second order multicritical point it suggests these models belong to distinct universality classes which differ from that of self attracting polymer chains which do not undergo a collapse theta transition on this fractal lattice

a transfer matrix approach to random resistor networks introduces a transfer matrix formulation to compute the conductance of random resistor networks the authors apply this method to strips of width up to 40 and use finite size scaling arguments to obtain an estimate for the conductivity critical exponent in two dimensions t 1 28 or 0 03

the dalgarno lewis method as a perturbation theory the dalgarno lewis summation technique as used in the rayleigh schrodinger perturbation theory is examined it is shown that this technique forms an independent perturbation theory and can be used to deal with both bound and continuum state problems with some added advantage over other such approaches

temporal development of invasion percolation the authors consider the dynamics of invasion percolation for different models this process exhibits scaling behaviours which in the simplest case can be related to some critical static properties of usual percolation in particular the concept of bursts is shown to be of fundamental importance the critical exponents they derive are checked numerically on a case of invasion percolation where growth is restricted to the hull of the existing cluster

beliefs about beliefs without probabilities this paper constructs a space of states of the world representing the exhaustive uncertainty facing each player in a strategic situation the innovation is that preferences are restricted primarily by regularity conditions and need not conform with subjective expected utility theory the construction employs a hierarchy of preferences rather than of beliefs as in the standard bayesian model the framework is sufficiently general to accommodate uncertainty averse preferences such as exhibited in the ellsberg paradox and to allow common knowledge of expected utility or choquet expected utility to be well defined formally applications include the provision of i foundations for a harsanyi style game of incomplete information and ii a rich framework for the axiomatization of solution concepts for complete information normal form games

speculation on a scaling law for superconductor resistor mixture exponent s in a percolation system it is argued that looking for a relationship between the exponent s and the cluster exponents may be more promising than looking for one between the resistor insulator exponent t and the cluster exponents the author proposes s nu beta 2

rationality computability and nash equilibrium suppose two agents play a game each using a computable algorithm to decide what to do these algorithms being common knowledge we show that it is possible to act rationally provided we limit our attention to a natural subset of solvable games and to opponents who use rational algorithms the outcome is a nash equilibrium going further we show that rationality is possible on many domains of games and opposing algorithms but each domain requires a particular solution algorithm no one algorithm is rational on all possible domains

universal amplitudes ratio at the collapse transition of polymers in two dimensions first values of the universal ratio of the mean square radius of gyration and the mean square end to end distance for 2d linear polymers at their collapse transition are calculated by the scanning simulation method two models are considered i self avoiding walks saws with nearest neighbour attractions for which this ratio is found to be 0 179 or 0 003 at the theta point ii self intersecting trails at their tricritical point for which the value 0 166 or 0 002 is obtained this provides yet another indication that these models may belong to different tricritical universality classes the authors have also computed this ratio for trails at the high temperature limit with non interacting intersections and found 0 141 or 0 001 in excellent agreement with earlier estimates of this ratio for saws in their swollen phase

transfer matrix calculation of the relative noise exponent in a two dimensional percolating network a new method is given for calculating the relative noise exponent describing the size dependence of macroscopic fluctuations due to microscopic resistance noise of a random resistor network at the percolation threshold pc the macroscopic fluctuation is calculated exactly using the transfer matrix idea in a given network the asymptotic exponent b is found by finite size scaling

ac properties of 2d percolation networks a transfer matrix approach the frequency dependent ac properties of 2d random resistor capacitor mixtures are studied through a numerical transfer matrix method the algorithm which generalises to the complex impedances used by herrmann et al 1984 permits a detailed study of the scaling laws obeyed by the complex dielectric constant in the critical regime low frequency and concentration close to the geometrical percolation threshold the duality property of the system is extensively used throughout the analysis quantities of experimental interest such as the loss angle delta and the cole and cole plot of the dielectric constant are also examined both close to and away from the critical point

resistances between two nodes of a path network abstract considerable progress has recently been made in the development of techniques to exactly determine two point resistances in networks of various topologies given n positive integers m 1 m 2 m n let p m i 1 n be the network with node set v v 1 v 2 v n where v i v j if i j and v i m i and with edge set e u v u v i v v i 1 i 1 2 n 1 in this paper using triangle star transformation and the principle of substitution we express arbitrary two nodes resistances of a path network in terms of the number of their nodes

discrete network models for the low field hall effect near a percolation threshold theory and simulations the critical behavior of the weak field hall effect near a percolation threshold is studied with the help of two discrete random network models many finite realizations of such networks at the percolation threshold are produced and solved to yield the potentials at all sites a new algorithm for doing that was developed that is based on the transfer matrix method the site potentials are used to calculate the bulk effective hall conductivity and hall coefficient as well as some other properties such as the ohmic conductivity the size of the backbone and the number of binodes scaling behavior for these quantities as power laws of the network size is determined and values of the critical exponents are found

current distribution on a three dimensional bond diluted random resistor network at the percolation threshold current and logarithm current distributions on a three dimensional random bond percolation cubic network were studied at the percolation threshold by computer simulations predictions of a hierarchical model that combine fractal structure and randomness agree with our numerical simulations in the thermodynamic limit the logarithm current distribution exhibits ann ln i i1 3 dependence below some characteristic currentic this distribution may scale with lni lnl but the data are insufficient to make this a definite conclusion due to the small range of lnl considered a study of the moments does not reveal this behavior and a study of the distribution itself is required

phenomenological renormalisation of monte carlo data for percolation the accuracy of a phenomenological renormalisation which is based on monte carlo data is tested by investigating site percolation in a simple cubic lattice the method appears to be very accurate and can yield precise estimates of the quantities of interest with small to moderate lattice sizes the site percolation threshold of the lattice is predicted to be 0 3115 or 0 0005 if nu is the exponent of correlation length the author finds beta nu approximately 0 48 or 0 01 for the exponent of percolation probability and beta b nu approximately 1 1 or 0 03 for the backbone exponent in agreement with most accurate data currently available an analysis of the data at the percolation threshold based on the standard finite size scaling method also supports the results

approximate shape invariant potentials in quantum mechanics we introduce and explore the concept of approximate shape invariance of parametrized potential functions for non relativistic one dimensional quantum systems the supersymmetric partner v2 of an approximate shape invariant potential v1 does not exhibit the same shape as the original potential however by an appropriate choice of the values of the relevant parameters v2 can still be approximated by v1 we also propose a measure for the degree of shape invariance exhibited by a parametrized potential function in order to illustrate these ideas we consider a the lillo mantegna potentials admitting exact analytic power law ground state wavefunctions and b a family of potentials whose ground state eigenfunctions are given by the trial wavefunctions employed by cooper dawson and shepard in the susy based variational method

a novel determination of the percolation exponent u in three dimensions the authors present an accurate determination of the percolation transport exponent u t s t in three dimensions based on a transfer matrix approach to the ac frequency dependent conductivity and a finite size scaling analysis of the numerical data the phase of the complex conductivity the loss angle assumes the universal value delta c pi 2 1 u at low frequency at the percolation threshold as a test of their numerical scheme the two dimensional exact duality result delta c pi 4 i e s t is recovered with a very good accuracy in three dimensions their data are extrapolated to tan delta c 0 54 or 0 03 i e u 0 69 or 0 02 whereas the usually accepted values s v 0 85 or 0 04 t v 2 2 or 0 1 yield u 0 72 or 0 02 this marginal disagreement can be attributed to ill behaved corrections to finite size scaling

percolation in restricted geometries and conformal invariance the authors determine numerically the pair connectivity for two dimensional percolation systems subject to different boundary geometries their results provide a sensitive and efficient test of the concept of conformal invariance and confirm its validity even for relatively small lattices the influence of boundaries on electrical conduction in percolative networks is also addressed briefly

how to decide how to decide how to modeling limited rationality it seems inconsistent to model boundedly rational action choice by assuming that the agent chooses the optimal decision procedure this criticism is not avoided by assuming that he chooses the optimal procedure to choose a procedure to to choose an action the author shows that properly interpreted this regress continued transfinitely generates a model representing the agent s perception of all his options including every way to refine his perceptions in this model the agent surely must choose the perceived best option hence it is not inconsistent to model limited rationality by assuming that the agent uses the optimal decision procedure copyright 1991 by the econometric society

on correction to scaling for two and three dimensional scalar and vector percolation a reanalysis of the resistancer of two and three dimensional superconducting networks at the percolation thresholdp c together with the previous results for the elastic modulik of such networks shows that there is aunified description of finite size scaling for scalar and vector transport properties of percolation systems for a network of linear sizel atp c and forboth scalar and vector percolation inboth two and three dimensions k andr scale withl asl x a 1 a 2 lnl 1 a 3 l 1 wherex is the ratio of the associated critical exponent ofk orr and the correlation length exponentv of percolation although our estimates ofx for the resistance of percolation networks are consistent with the previous results they do indicate that inboth two and three dimensions and forboth scalar and vector percolation the leading nonanalytic correction to scaling exponent iszero from a reanalysis of data on diffusion on percolation clusters atp c we propose that such correction to scaling terms are a general property ofdynamics of percolation clusters we also suggest that for two dimensional percolation the conductivity exponentt and the superconductivity exponents are given by s t v 4 187 144 1 2986 and the elasticity exponentf is given byf t 2v 571 144 3 9652 where is the exponent of the strength of the infinite percolation cluster

universal amplitude exponent relations for interfaces and conformal mappings the author considers the finite size fs scaling behaviour of the transverse correlation lengths xi sub n for an interface confined in a strip geometry exactly at a second order fluctuation dominated wetting transition he shows that the dimensionless amplitude ratios a sub n sigma l2 kbt xi sub n with sigma the surface stiffness coefficient are related to the short distance expansion critical exponent theta by a sub n pi 2n n theta 1 2 for all fluctuation dominated wetting transitions provided the binding potential v y in the capillary wave model is conformally mapped from the semi infinite plane he argues that the existence of this amplitude exponent relation reflects the conformal invariance of the capillary wave model lagrange density and lists a number of one point functions which can be shown to exhibit this invariance explicitly

new method for the exact determination of the effective conductivity and the local field in rlc networks we present a new numerical method for exactly determining the effective conductivity and the local field for random rlc networks this method is compared with a real space renormalization group method and the frank and lobb fl method although our method is slower than the fl method it also exactly computes the local field for large size systems we also show that the renormalization group method fails in determining the local field

on directed interacting animals and directed percolation we study the phase diagram of fully directed lattice animals with nearest neighbour interactions on the square lattice this model comprises several interesting ensembles directed site and bond trees bond animals strongly embeddable animals as special cases and its collapse transition is equivalent to a directed bond percolation threshold precise estimates for the animal size exponents in the different phases and for the critical fugacities of these special ensembles are obtained from a phenomenological renormalization group analysis of the correlation lengths for strips of width up to n 17 the crossover region in the vicinity of the collapse transition is analysed in detail and the crossover exponent is determined directly from the singular part of the free energy we show using scaling arguments and an exact relation due to dhar that is equal to the fisher exponent governing the size distribution of large directed percolation clusters

numerical study of transport properties in continuum percolation the authors present numerical simulations of ac conductance for a random resistor capacitor network the conductance obeys a probability density function p g varies as g alpha 0 alpha 1 they use a highly efficient propagation algorithm to calculate the effective conductance of a long strip of a lattice at low frequencies they find that for the concentration p of conducting bonds less than the percolation threshold pc the imaginary part of conductance is proportional to frequency im geff approximately omega and the real part of conductance shows an anomalous frequency dependence re geff approximately omega 2 alpha the results of simulations in such a continuum system are in agreement with the predictions of the effective medium and the maxwell garnett approximation they also calculate the non universal dc conductivity exponents in continuum percolation the results are consistent with earlier theoretical predictions and numerical calculations

distribution of shortest path lengths in subcritical erdős renyi networks networks that are fragmented into small disconnected components are prevalent in a large variety of systems these include the secure communication networks of commercial enterprises government agencies and illicit organizations as well as networks that suffered multiple failures attacks or epidemics the structural and statistical properties of such networks resemble those of subcritical random networks which consist of finite components whose sizes are nonextensive surprisingly such networks do not exhibit the small world property that is typical in supercritical random networks where the mean distance between pairs of nodes scales logarithmically with the network size unlike supercritical networks whose structure has been studied extensively subcritical networks have attracted relatively little attention a special feature of these networks is that the statistical and geometric properties vary between different components and depend on their sizes and topologies the overall statistics of the network can be obtained by a summation over all the components with suitable weights we use a topological expansion to perform a systematic analysis of the degree distribution and the distribution of shortest path lengths dspl on components of given sizes and topologies in subcritical erd ifmmode mbox h o else h o fi s r enyi er networks from this expansion we obtain an exact analytical expression for the dspl of the entire subcritical network in the asymptotic limit the dspl which accounts for all the pairs of nodes that reside on the same finite component fc is found to follow a geometric distribution of the form p mathrm fc l ensuremath ell ll ensuremath infty 1 ensuremath c c ensuremath ell ensuremath 1 where cl1 is the mean degree using computer simulations we calculate the dspl in subcritical er networks of increasing sizes and confirm the convergence to this asymptotic result we also obtain exact asymptotic results for the mean distance ensuremath langle l ensuremath rangle mathrm fc and for the standard deviation of the dspl ensuremath sigma l mathrm fc and show that the simulation results converge to these asymptotic results using the duality relations between subcritical and supercritical er networks we obtain the dspl on the nongiant components of er networks above the percolation transition

on renormalisation of fluctuations at the onset of a centre manifold the author considers a dissipative reactive system driven far from thermodynamic equilibrium and sustaining a locally attractive centre manifold beyond a symmetry breaking instability the author derives a functional equation for the renormalised diffusion pressure this coefficient gives autocorrelations for intrinsic fluctuations and depends on the bifurcation parameter the system is subject to the following realistic restrictions i the temporal organisation lying beyond the instability can be accounted for by the existence of a centre manifold ii the effective equilibrium energy averaged with respect to the stationary distribution about the centre manifold is independent of the bifurcation parameter

a search for shape invariant solvable potentials the author investigates a simple method of constructing potentials which is related to the work of bhattacharjie and sudarshan 1962 and for which the schrodinger equation can be solved in terms of known special functions it turns out that this method can be related to supersymmetric quantum mechanics and this relationship can help to decide which special functions satisfying linear homogeneous second order differential equations can be solutions of the schrodinger equation with potentials of the form v x w2 x w x the author illustrates this procedure with the example of orthogonal polynomials and obtains explicit expressions of wavefunctions of a wide class of shape invariant potentials

an algebraic approach to problems with polynomial hamiltonians on euclidean spaces explicit expressions are given for the actions and radial matrix elements of basic radial observables on multi dimensional spaces in a continuous sequence of orthonormal bases for unitary su 1 1 irreps explicit expressions are also given for so n reduced matrix elements of basic orbital observables these developments make it possible to determine the matrix elements of polynomial and other hamiltonians analytically to within so n clebsch gordan coefficients and to select an optimal basis for a particular problem such that the expansion of eigenfunctions is most rapidly convergent

pt symmetric potentials and the so 2 2 algebra starting from a differential realization of the generators of the so 2 2 algebra we connect the eigenvalue equation of the casimir invariant either with the hypergeometric equation or the schr odinger equation in the latter case we consider problems for whichso 2 2 appears as a potentia la lgebra connecting states with the same energy in different potentials we analyse the role of the two so 2 1 subalgebras and point out their importance for pt symmetric problems where the doubling of bound states is known to occur we present two mechanisms for this and illustrate them with the example of the scarf and the p oschl teller ii potentials we also analyse scattering states transmission

a real space renormalisation group approach to electrical and noise properties of percolation clusters the migdal kadanoff renormalisation scheme which becomes exact on hierarchical lattices is used to investigate electrical properties of 2d and 3d percolation clusters the author studies in detail the frequency dependence of the conductivity and the loss angle and the amplification of resistance flicker noise this type of noise exhibits critical amplification with its own exponents x and y which are related to the exponent b recently defined by rammal et al 1985

the resistive susceptibility of random diode insulator networks the authors consider regular crystal lattices in which the bonds are either diodes probability p or insulators the mean number of backbone bonds lbb p when averaged over lattice points is found to be related to the mean size of clusters s p by lbb p zps p where z is the number of bonds directed away from any lattice site thus lbb diverges at pc with the mean size exponent gamma the resistive susceptibility chi r p of harris and fisch 1977 1978 is expanded graphically as a power series in p and it is found that term by term chi r p psi r p s p 2 where psi r p is obtained from chi r p by ignoring contributions from nodal graphs the above results are valid for bond and site dilution in any dimension for bond dilution on the square lattice the authors have determined psi r p to p17 and pade analysis of the resulting series for chi r p shows that it diverges with exponent gamma r 3 654 or 0 017 using a scaling relation the exponent t for the conductivity of the infinite cluster is estimated to be t 0 75 or 0 02

the backbone and conductivity of random clusters the author studies the properties of the backbone of and conduction in random clusters which have homogeneous interior structure these include the largest percolation cluster at pc lattice animals and the witten sander ws aggregates for lattice animals the fractal dimension da of the backbone is estimated the for first time and is found to be about 1 14 at d 2 and about 1 39 at d 3 these values are much lower than the corresponding values for the animals themselves the fractal and spectral dimensions of the backbone of the ws aggregates are estimated to be about 1 25 and 1 06 at d 2 respectively a recent hypothesis which relates the fractal dimension of random walks on these clusters to those of the clusters and their backbones is also discussed it is shown that for the ws aggregates ds ds 2 on a bethe lattice i e at d infinity in contrast with ds 4 3 and ds 1 for percolation clusters and lattice animals the conductivity exponent ta of lattice animals is found to be ta d 2 approximately 0 73 and ta d 3 approximately 1 19 and ta d or 8 2 whereas for the ws aggregates the author finds tws d 2 approximately 0 67 and tws d 3 approximately 0 94 and tws d to infinity 1

surface growth on percolation networks by a conserved noise restricted solid on solid growth model surface growth by the conserved noise restricted solid on solid model is investigated on diluted lattices i e on percolation networks that are embedded in two spatial dimensions the growth exponent and the roughness exponent are defined respectively by the mean square surface width via w 2 t t 2 and the mean square saturated width via w sat 2 l l 2 where l is the system size these are measured on both an infinite network and a backbone network and the results are compared with power counting predictions obtained using the fractional langevin equation while the monte carlo results on deterministic fractal substrates show excellent agreement with the predictions d h kim and j m kim phys rev e 84 011105 2011 the results on critical percolation networks deviate by 8 12 from these predictions

the transport exponent in percolation models with additional loops abstract several percolation models with additional loops were studied the transport exponents for these models were estimated numerically by means of a transfer matrix approach it was found that the transport exponent has a drastically changed value for some of the models this result supports some previous numerical studies on the vibrational properties of similar models with additional loops

electrical conductivity of finite size percolation networks calculating by a monte carlo technique the conductivities of small two dimensional square bond percolation networks 4 4 to 50 50 at a limited number of probability values from the critical region to full conductivity the authors have been able to show by an original interpolation technique that in the critical region the conductivity function does indeed obey a universal fisher type finite size scaling function they further show that this technique permits them to deduce values of the critical exponents for both the conductivity and the correlation length even from calculations on relatively small networks