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

multiscaling in discrete models of epitaxial growth the nature of the approximate multi exponent scaling of height fluctuations found in a class of limited mobility atomistic models of epitaxial growth is investigated numerically a recently proposed connection between this multiscaling behavior and a controlled instability in discretized versions of a conserved growth equation is elucidated

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

compact moduli spaces of del pezzo surfaces and kahler einstein metrics we prove that the gromov hausdorff compactification of the moduli space of kahler einstein del pezzo surfaces in each degree agrees with certain algebro geometric compactification in particular this recovers tian s theorem on the existence of kahler einstein metrics on smooth del pezzo surfaces and classifies all the degenerations of such metrics the proof is based on a combination of both algebraic and differential geometric techniques

instability intermittency and multiscaling in discrete growth models of kinetic roughening we show by numerical simulations that discretized versions of commonly studied continuum nonlinear growth equations such as the kardar parisi zhangequation and the lai das sarma villain equation and related atomistic models of epitaxial growth have a generic instability in which isolated pillars or grooves on an otherwise flat interface grow in time when their height or depth exceeds a critical value depending on the details of the model the instability found in the discretized version may or may not be present in the truly continuum growth equation indicating that the behavior of discretized nonlinear growth equations may be very different from that of their continuum counterparts this instability can be controlled either by the introduction of higher order nonlinear terms with appropriate coefficients or by restricting the growth of pillars or grooves by other means a number of such controlled instability models are studied by simulation for appropriate choice of the parameters used for controlling the instability these models exhibit intermittent behavior characterized by multiexponent scaling of height fluctuations over the time interval during which the instability is active the behavior found in this regime is very similar to the turbulent behavior observed in recent simulations of several one and two dimensional atomistic models of epitaxial growth

on automorphisms of enriques surfaces an enriques surface over an algebraically closed field k of characteristic 4 2 is a nonsingular projective surface f with hi f gv h2 f or 0 2kv 0 the unramified double cover of f defined by the torsion class k v is a k3 surface f a nonsingular projective surface with hi f gr 0 kr 0 the study of enriques surfaces is equivalent to the study of k3 surfaces with a fixed point free involution z in particular the automor phism group aut f of f is isomorphic to the group aut ff z z where aut f z is the centralizer of z in the automorphism group aut f of ft in the case k the field of complex numbers the study of aut f is based on the global torelli theorem for k3 surfaces proven by i piatetski shapiro and i shafarevich in 19 it follows from this theorem that up to a finite group the group aut ff is isomorphic to the quotient group o pic f w where o pic f is the orthogonal group of the picard lattice of ff and w its normal subgroup generated by the reflections into the classes of nonsingular rational curves for a generic enriques surface f this theorem allows to compute aut f see 3 and also 17 where this result is not stated explicitly for an arbitrary f the relation between f and ff does not help since it is very difficult to compute the action of z in pie if however by other means we can prove the following analog of piatetski shapiro and shafarevich s result

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

brill noether theory of curves on enriques surfaces i the positive cone and gonality we study the existence of linear series on curves lying on an enriques surface and general in their complete linear system using a method that works also below the bogomolov reider range we compute in all cases the gonality of such curves we also give a new result about the positive cone of line bundles on an enriques surface and we show how this relates to the gonality

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

scalar curvature and stability of toric varieties we define a stability condition for a polarised algebraic variety and state a conjecture relating this to the existence of a kahler metric of constant scalar curvature the main result of the paper goes some way towards verifying this conjecture in the case of toric surfaces we prove that under the stability hypothesis the mabuchi functional is bounded below on invariant metrics and that minimising sequences have a certain convergence property in the reverse direction we give new examples of polarised surfaces which do not admit metrics of constant scalar curvature the proofs use a general framework developed by guillemin and abreu in which invariant kahler metrics correspond to convex functions on the moment polytope of a toric variety this paper is a step towards the solution of the general problem of finding conditions under which a complex projective variety admits a kahler metric of constant scalar curvature the pattern of the answer one expects is that this differential geometric condition should be equivalent to some notion of stability in the sense of geometric invariant theory this expectation is probably now an item of folklore going back to suggestions put forward by yau in the case of kahlereinstein metrics and the many results of tian and others in this case reinforced by a detailed formal picture which makes clear the analogy with the well established relation between the stability of vector bundles and yang mills connections 5 here we begin the investigation of

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

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

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

failure of the hasse principle for enriques surfaces abstract we construct an enriques surface x over q with empty etale brauer set and hence no rational points for which there is no algebraic brauer manin obstruction to the hasse principle in addition if there is a transcendental obstruction on x then we obtain a k3 surface that has a transcendental obstruction to the hasse principle

persistence in nonequilibrium surface growth persistence probabilities of the interface height in 1 1 and 2 1 dimensional atomistic solid on solid stochastic models of surface growth are studied using kinetic monte carlo simulations with emphasis on models that belong to the molecular beam epitaxy mbe universality class both the initial transient and the long time steady state regimes are investigated we show that for growth models in the mbe universality class the nonlinearity of the underlying dynamical equation is clearly reflected in the difference between the measured values of the positive and negative persistence exponents in both transient and steady state regimes for the mbe universality class the positive and negative persistence exponents in the steady state are found to be 0 66 0 02 and 0 78 0 02 respectively in 1 1 dimensions and 0 76 0 02 and 0 85 0 02 respectively in 2 1 dimensions the noise reduction technique is applied on some of the 1 1 dimensional models in order to obtain accurate values of the persistence exponents we show analytically that a relation between the steady state persistence exponent and the dynamic growth exponent found earlier to be valid for linear models should be satisfied by the smaller of the two steady state persistence exponents in the nonlinear models our numerical results for the persistence exponents are consistent with this prediction we also find that the steady state persistence exponents can be obtained from simulations over times that are much shorter than that required for the interface to reach the steady state the dependence of the persistence probability on the system size and the sampling time is shown to be described by a simple scaling form

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

equations of log del pezzo surfaces of index 2 del pezzo surfaces over mathbb c with log terminal singularities of index 2 were classified by alekseev and nikulin in this paper for each of these surfaces we find an appropriate morphism to projective space these morphisms enable us to describe equations of natural embeddings of log del pezzo surfaces of index 2 in some weighted projective space the results obtained give a completion of similar results of du val hidaka and watanabe describing del pezzo surfaces of index 1

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

extended self similarity in kinetic surface roughening we show from simulations that a limited mobility solid on solid model of kinetically rough surface growth exhibits extended self similarity analogous to that found in fluid turbulence the range over which scale independent power law behavior is observed is significantly enhanced if two correlation functions of different order such as those representing two different moments of the difference in height between two points are plotted against each other this behavior found in both one and two dimensions suggests that the relative exponents may be more fundamental than the absolute ones

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

on irregular threefolds and fourfolds with numerically trivial canonical bundle we prove that for a smooth projective irregular threefold x with k x equiv 0 and a nef and big divisor l on x ml p gives a birational map for all m ge 3 and all p in mathrm pic 0 x we also use the same method to deal with fourfolds and prove that for a smooth projective irregular fourfold x with k x equiv 0 and an ample divisor l on x ml p gives a birational map for all m ge 5 and all p in mathrm pic 0 x these results are optimal

automorphisms of enriques surfaces o introduction the aim of this note is to compute the group aut y of biholomorphic auto morphisms for the general enriques surface y the basic tool is the global to relli theorem for projective k3 surfaces as it was given by piatetski shapiro and shafarevich 11 and refined by burns and rapaport 2 the essential result is that in contrast to the case of curves aut y is big for general y and small for special y in this paper we consider the complex case only recall that an enriques surface y is a projective complex surface with universal double cover a k3 surface one knows that h2 y z 7z 2 2 and that the cup product provides

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 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

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

on positivity of line bundles on enriques surfaces we study linear systems on enriques surfaces we prove rationality of seshadri constants of ample line bundles on enriques surfaces and provide lower bounds on these numbers we show the nonexistence of k very ample line bundles on enriques surfaces of degree 4k 4 for k 1 thus answering an old question of ballico and sommese introduction in this note we answer a question posed by ballico and sommese 1 see also terakawa 14 concerning the existence of k very ample line bundles of degree 4k 4 on enriques surfaces we show that such line bundles exist if and only if k 0 and that in fact the degree of a k very ample line bundle on an enriques surface is subject to a much more restrictive bound theorem 2 4 this has strong consequences for the local positivity of line bundles on enriques surfaces we address this questions in section 3 in particular we show that seshadri constants of ample line bundles on enriques surfaces are always rational 1 auxiliary material and notation the basic reference for this paper is the monograph 6 by cossec and dolgachev here we briefly recall properties of enriques surfaces needed for our considerations first of all an enriques surface is a surface y of kodaira dimension zero with no differential forms in particular ho ky 0 and the canonical bundle ky is a 2torsion element in pico y i e 2ky oy to alleviate notation it is convenient to use the following convention in the sequel we make frequent use of vanishing theorems and statements on adjoint line bundles since the canonical divisor ky is numerically trivial the adjoint line bundle ky l has the same numerical properties as l in particular if l is nef and big ample then so is ky l by the nakai moishezon criterion so in order to get vanishing statements for l we apply vanishing theorems to ky l and use the fact that 2ky is trivial in the sequel we do so without comment and hope to cause no confusion this way received by the editors june 1 2000 2000 mathematics subject classification primary 14j28 secondary 14c20 14e25

mound formation and coarsening from a nonlinear instability in surface growth we study spatially discretized versions of a class of one dimensional nonequilibrium conserved growth equations for both nonconserved and conserved noise using numerical integration an atomistic version of these growth equations is also studied using stochastic simulation the models with nonconserved noise are found to exhibit mound formation and power law coarsening with slope selection for a range of values of the model parameters unlike previously proposed models of mound formation the ehrlich schwoebel step edge barrier usually modeled as a linear instability in growth equations is absent in our models mound formation in our models occurs due to a nonlinear instability in which the height depth of spontaneously generated pillars grooves increases rapidly if the initial height depth is sufficiently large when this instability is controlled by the introduction of a nonlinear control function the system exhibits a first order dynamical phase transition from a rough self affine phase to a mounded one as the value of the parameter that measures the effectiveness of control is decreased we define an order parameter that may be used to distinguish between these two phases in the mounded phase the system exhibits power law coarsening of the mounds in which a selected slope is retained at all times the coarsening exponents for the spatially discretized continuum equation and the atomistic model are found to be different an explanation of this difference is proposed and verified by simulations in the spatially discretized growth equation with conserved noise we find the curious result that the kinetically rough and mounded phases are both locally stable in a region of parameter space in this region the initial configuration of the system determines its steady state behavior

kahler einstein metrics and the generalized futaki invariant in 1983 futaki introduced his famous invariant this invariant generalizes the obstruction of kazdan warner to prescribing gauss curvature on s 2 cf ful l the futaki invariant is defined for any compact k ihler manifold with positive first chern class that has nontrivial holomorphic vector fields it is a lie algebraic character from the lie algebra of holomorphic vector fields into and its vanishing is a necessary condition for the existence of a k ihler einstein metric on the underlying manifold therefore it can be used to test the existence of k ihlereinstein metrics on a given compact k ihler manifold with positive first chern class an excellent reference on the futaki invariant is futaki s book fu2 until now all known nontrivial obstructions to k ihler einstein metrics come from holomorphic vector fields this suggests the following conjecture

ricci curvature bounds and einstein metrics on compact manifolds where f mo m1 is a homeomorphism and dil f is the dilatation of f given by dil f supx x2 dist f x f x2 dist x1 x2 if mo and m1 are not homeomorphic define dl mo mi oo gromov 20 proves the remarkable result that the space of compact riemannian manifolds f a 3 d of sectional curvature iki 3 0 and diameter dm v and diameter dm c iki dm vm1 in particular gromov s compactness theorem may be strengthened to the statement that f a v d is c1 l compact in the lipschitz topology in this paper we study the question of lipschitz convergence of compact riemannian manifolds with bounds imposed on the ricci curvature ric in

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

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

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

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

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

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

on the picard group of enriques surfaces an enriques surface over an algebraically closed field k of characteristic 4 2 is a non singular projective surface s with h s d s h2 s cs 0 and 2ks 0 the unramitied double cover defined by ks is a k3 surface r a non singular projective surface with hi r d r 0 kr 0 illusie has shown i that the group of divisors modulo numerical equivalence is isomorphic to the enriques lattice u e 8 1 where u and e s 1 denote respectively the unique even unimodular lattices of index of inertia 1 1 and 0 8 the purpose of this note is to use this isomorphism to study the picard group of s we prove the existence of certain configurations of irreducible curves of arithmetic genus 0 or 1 and deduce from them the existence of certain projective models for s and r for example we prove the following results

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

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

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

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

moduli of curves on enriques surfaces abstract we compute the number of moduli of all irreducible components of the moduli space of smooth curves on enriques surfaces in most cases the moduli maps to the moduli space of prym curves are generically injective or dominant exceptional behavior is related to existence of enriques fano threefolds and to curves with nodal prym canonical model

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

controlled instability and multiscaling in models of epitaxial growth we show that discretized versions of commonly studied nonlinear growth equations have a generic instability in which isolated pillars or grooves on an otherwise flat interface grow in time when their height or depth exceeds a critical value controlling this instability by the introduction of higher order nonlinear terms leads io intermittent behavior characterized by multiexponent scaling of height fluctuations similar to the turbulent behavior found in recent simulations of one dimensional atomistic models of epitaxial growth

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