on the distribution of a random variable occurring in 1d disordered systems the authors consider the random variable z 1 x1 x1x2 x1x2x3 where the xi are independent identically distributed variables they derive some asymptotic properties of the distribution of z which are related e g to the low temperature behaviour of the random field ising chain for a special class of distributions of the xi exact solutions are presented they also study the cases where the distribution function of z exhibits a power law fall off modulated by a periodic critical amplitude

critical properties of random site percolation in two and three dimensions a monte carlo study abstract we measure the critical exponents of two dimensional and three dimensional random site percolation and find excellent agreement with nienhuis exact results two dimensions and good agreement with other numerical work three dimensions we also measure the correlation length amplitude ratio and the mean cluster size amplitude ratio in two and three dimensions we report values of 4 0 0 5 2 d 2 0 0 5 3 d and 75 40 26 2 d and 8 3 2 3 d for the correlation length and the mean cluster size amplitude ratios respectively a direct measurement is made of the fractal dimension d f at the critical point p c 0 50 of the triangular lattice d f is 1 90 0 01 while at p c 0 3117 for the simple cubic lattice d f is 2 50 0 02

flory calculation of the fractal dimensionality of the shortest path in a percolation cluster a flory type calculation of the fractal dimensionality of the shortest path in a percolation cluster near the threshold is introduced the result is roughly consistent with previous predictions or numerical simulations

reciprocal relations for the weight factors arising in the series expansion of backbone percolation functions the authors consider several properties of the backbone of percolation clusters the percolation model on a graph g is defined as follows each element vertex or edge of g is either open or closed the element e being open with probability pe independently of all other elements the u v backbone in any configuration is the subgraph consisting of all elements which belong to at least one open self avoiding path from a vertex u to a vertex v they show that the expected value of any u v backbone random variable z which depends only on the elements of the u v backbone e g z may be the number of edges or the length of the shortest path of the backbone is given by e z sigma k contained in p 1 mod k mod 1zk pi e in g k pe where p is the set of all self avoiding paths from u to v on g g k is the subgraph formed by all elements which belong to at least one self avoiding path of k and the weightzk is given by zk sigma j contained in k 1 mod jmod 1 zj where zj is the value of z when the backbone is g j certain reciprocal relations are found to hold between the weights and the backbone variables in some important cases using these results series expansions for the various properties can be obtained

exact enumeration approach to fractal properties of the percolation backbone and 1 σ expansion an exact numeration approach is developed for the backbone fractal of the incipient infinite cluster at the percolation threshold the authors use this approach to calculate exactly the first low density expansion of lbb p for arbitrary system dimensionality d where lbb p is the mean of backbone bonds and p is the bond occupation probability standard series extrapolation methods provide estimates of the fractal dimension of the backbone for all d these disagree with the sierpinski gasket model of the backbone they also calculate the first low density expansions of lmin p and lred p which are respectively the mean number of bonds in the minimum path between i and j and the mean number of singly connected red bonds

exact enumeration of self avoiding walks on critical percolation clusters in 2 7 dimensions we study self avoiding walks on critical percolation clusters by means of a recently developed exact enumeration method which can handle walks of several thousand steps we had previously presented results for the two and three dimensional cases here we take a wider perspective and vary the system s dimensions up to d 7 beyond the supposed upper critical dimension of these results may serve to check analytical predictions and help understand how the medium s fractal structure impacts on the walks scaling behavior

epidemic models and percolation the authors argue that local epidemic models with immunisation are in the same universality class as percolation cluster growth models and show that the static exponents are equal to all orders in the epsilon expansion they calculate the dynamic exponent nu t 1 1 28 epsilon o epsilon 2 and relate this to other exponents involving the chemical distance

the fractal dimension of the minimum path in two and three dimensional percolation the authors calculate the fractal dimension d min of the shortest path l between two points on a percolation cluster where l approximately rd min and r is the pythagorean distance between the points they find d min 1 130 or 0 002 for d 2 and 1 34 or 0 01 for d 3

the chemical distance distribution in percolation clusters the authors study the conditional probability density p r mod l for the geometrical distance r corresponding to a given chemical distance l for percolation clusters in two dimensions they argue that i p r mod l alxg exp axdelta where x r lnu g 2 5 or 0 3 delta 9 8 or 0 5 and nu 0 88 or 0 02 and ii there is a relation delta 1 nu 1 which is in good agreement with numerical data these results are derived by considering a special class of self avoiding walks consisting of chains which are chemical paths shortest paths on critical percolation clusters

a note on coniglio s lemma it is proved that coniglio s 1981 1982 lemma holds for any pair of points in any graph in the low density series expansion of the cut edge weighted pair connectedness

invasion percolation new algorithms and universality classes employing highly efficient algorithms for simulating invasion percolation ip whose execution time scales as o mlog m or better for a cluster of m sites and for determining the backbone of the cluster we obtain precise estimates for the fractal dimensions of the sample spanning cluster the backbone and the minimal path in order to identify the universality classes of four different ip processes site and bond ip with and without trapping in two dimensions ip is characterized by two universality classes one each for ip without trapping and site and bond ip with trapping in a three dimensional site ip with and without trapping is in the universality class of random percolation while bond ip with trapping is in a distinct universality class which may be the same as that of optimal paths in strongly disordered media

red bond exponents of the critical and the tricritical ising model in three dimensions using the wolff and geometric cluster algorithms and finite size scaling analysis we investigate the critical ising and the tricritical blume capel models with nearest neighbor interactions on the simple cubic lattice the sampling procedure involves the decomposition of the ising configuration into geometric clusters each of which consists of a set of nearest neighboring spins of the same sign connected with bond probability p these clusters include the well known kasteleyn fortuin clusters as a special case for p 1 exp 2k where k is the ising spin spin coupling along the critical line k kc the size distribution of geometric clusters is investigated as a function of p we observe that unlike in the case of two dimensional tricriticality the percolation threshold in both models lies at pc 1 exp 2kc further we determine the corresponding red bond exponents as yr 0 757 2 and 0 501 5 for the critical ising and the tricritical blume capel models respectively on this basis we conjecture yr 1 2 for the latter model

theory of self avoiding walks on percolation fractals a phenomenological approach which takes into account the basic geometry and topology of percolation fractal structures and of self avoiding walks saw is used to derive a new expression for the flory exponent describing the average radius of gyration of saws on fractals we focus on the radius of gyration and discuss the importance of the intrinsic fractal dimensions of percolation clusters in determining the lower and upper critical dimensions of saws the mean field version of our new formula corresponds to the aharony and harris expression who used the standard flory approach for its derivation

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

the starry sky model of the percolation phase transition iii a new equilibrium equation and two kinds of the special elastic nodes abstract the equilibrium lame equation in two dimensional and three dimensional spaces can be presented as conductivity in magnetic field equation with lorenz force expression this allows us to apply the perturbation theory and generalize the lame equation and obtain a new equilibrium equation with all responses of compression and tension it means that compression in one direction leads to tension in the perpendicular direction and vice versa if the boundaries are fixed the solution in recurrent integral form for contributions to all orders of poisson ratio is then obtained zero order solution of lame equilibrium equation expresses the poisson ratio as ve p pc k k 0 3 and creates the special elastic nodes with maximum local deformation the first order solution gives mean values of effective lateral displacements or forces when lateral boundaries are fixed and expresses the bulk modulus as ke p pc 2t g where g 0 6 is a hall coefficient critical exponent it creates the special active elastic nodes where the whole sample microrotates around a new universality class of elasticity problem belongs to the hall coefficient universality class therefore the elasticity backbone coincides with the hall coefficient backbone with the fractal dimension d f b 2 25

a new percolation statistic with unusual properties in bond percolation on lattice with p pc there are unoccupied bonds which if occupied would immediately join the backbone in general when such a bond is occupied several tag end bonds may join the backbone as well the author defines ni to be the number of bonds that would join the backbone if such a bond i were occupied and n to be the average of ni taken over all such bonds on the lattice the author derives a formula for n and finds that for all lattices near pc n gamma bpc p pc 1 where gamma b is the exponent for the divergence of the backbone fraction identical formulae apply to site percolation

evidence for non universal exponents in bootstrap percolation bootstrap percolation bp models are systems where sites are initially randomly occupied those sites that do not maintain a suitable local environment of at least m occupied sites are successively removed current open questions concerning these systems now focus on the universality of critical exponents for the lower m values simulations indicate that for m 3 on the simple cubic lattice the exponent beta which characterizes the critical behaviour of the percolation probability is distinct from that of usual random percolation m 0 the authors quote pc 0 5717 or 0 0005 and beta 0 6 or 0 1 for m 3 they argue that the connectivity exponent nu for m 2 should be the same as that of usual random percolation and within the limits of numerical accuracy the authors observe that this appears to be the case for m 3 on the simple cubic lattice the implications of this surprising dichotomy are considered

eden growth on fractal media abstract simulations are presented for two dimensional eden growth into fractal networks we find that the fractal dimension of the eden cluster d e does not have a simple behavior when the cluster grows into the structure of a fractal network we also find that d e depends strongly on the minimal path between two arbitrary points in the network backbone and on the number of paths that appear between these two points at each scale

from rational bubbles to crashes we study and generalize in various ways the model of rational expectation re bubbles introduced by blanchard and watson in the economic literature bubbles are argued to be the equivalent of goldstone modes of the fundamental rational pricing equation associated with the symmetry breaking introduced by non vanishing dividends generalizing bubbles in terms of multiplicative stochastic maps we summarize the result of lux and sornette that the no arbitrage condition imposes that the tail of the return distribution is hyperbolic with an exponent 1 we then outline the main results of malevergne and sornette who extend the re bubble model to arbitrary dimensions d a number d of market time series are made linearly interdependent via d d stochastic coupling coefficients we derive the no arbitrage condition in this context and with the renewal theory for products of random matrices applied to stochastic recurrence equations we extend the theorem of lux and sornette to demonstrate that the tails of the unconditional distributions associated with such d dimensional bubble processes follow power laws with the same asymptotic tail exponent 1 for all assets the distribution of price differences and of returns is dominated by the same power law over an extended range of large returns although power law tails are a pervasive feature of empirical data the numerical value 1 is in disagreement with the usual empirical estimates 3 we then discuss two extensions the crash hazard rate model and the non stationary growth rate model of the re bubble model that provide two ways of reconciliation with the stylized facts of financial data

series study of random percolation in three dimensions new high density series data for the mean number percolation probability and susceptibility of finite clusters are presented for bond and site percolation on four standard three dimensional lattices a pade approximant analysis of both high and low density series makes particular use of the rather precise unbiased estimates of the percolation threshold pc obtained recently by heermann and stauffer 1981 for the simple cubic lattice for this lattice the authors obtain the biased estimate gamma 1 73 or 0 03 for the site problem and a similar estimate but with larger uncertainties for the bond problem such a value is significantly larger than earlier series estimates assuming gamma to be universal they obtain precise although biased estimates of pc for both bond and site percolation on all four lattices using the bond estimates of pc they find an overall biased estimate of beta 0 454 or 0 008 for bond percolation on all three dimensional lattices the corresponding site problem requires further study scaling estimates of other critical exponents are alpha 0 64 or 0 05 delta 4 81 or 0 14 delta 2 18 or 0 04 nu 0 88 or 0 02 and nu 0 03 or 0 03

backbone and elastic backbone of percolation clusters obtained by the new method of burning we introduce a method of obtaining information about percolation clusters in which the clusters are burned in different ways this method yields the backbone and the elastic backbone their number of loops shortest path and other quantities it works in any dimension and can be applied also to other clusters we obtain accurate estimates for the backbone exponents in two and three dimensions

spreading of percolation in three and four dimensions the authors present monte carlo simulations of the spreading of bond percolation on three and four dimensional simple hyper cubic lattices the algorithm is the same as that applied previously in two dimensions with the spreading proceeding from a hyperplane they find the spreading dimension to be d 1 82 or 0 02 three dimensional and d 1 88 or 0 03 four dimensional also they obtain values of the percolation probability and of the static exponents with errors at least comparable to the best values from the literature

backbone exponents of the two dimensional q state potts model a monte carlo investigation we determine the backbone exponent x b of several critical and tricritical q state potts models in two dimensions the critical systems include the bond percolation the ising the q 2 sqrt 3 3 and 4 state potts and the baxter wu model and the tricritical ones include the q 1 potts model and the blume capel model for this purpose we formulate several efficient monte carlo methods and sample the probability p2 of a pair of points connected via at least two independent paths finite size scaling analysis of p2 yields x b as 0 3566 2 0 2696 3 0 2105 3 and 0 127 4 for the critical q 2 sqrt 3 1 2 3 and 4 state potts model respectively at tricriticality we obtain x b 0 0520 3 and 0 0753 6 for the q 1 and 2 potts model respectively for the critical q 0 potts model it is derived that x b 3 4 from a scaling argument we find that at tricriticality x b reduces to the magnetic exponent as confirmed by the numerical results

topological loop intermittency in the three dimensional turbulence a new topological or loop type of intermittency is studied in three dimensional turbulence the spectral or fracton dimension ds 7 4 7 1 95 for this turbulence i e all states are localized this intermittency is different both from kolmogorov type unlocalized since ds 2 and from alexander orbach type strong localized helical fractons with ds 43 the corresponding energy spectral exponent is obtained to be equal to 1 7 1 65 these results are in good agreement with recent numerical spectral dimension and laboratory turbulent energy spectrum data the anomaly in the experimentally observed fractal dimension of clouds surface d 2 35 can be also explained in the framework of this intermittency phenomenon the results have been obtained by a renormalization of the noise like 1k energy spectrum in isotropic incompressible turbulence with taking into account of spontaneous parity breaking by virtual pairs of helical excitations

linear stochastic dynamics with nonlinear fractal properties abstract stochastic processes with multiplicative noise have been studied independently in several different contexts over the past decades we focus on the regime found for a generic set of control parameters in which stochastic processes with multiplicative noise produce intermittency of a special kind characterized by a power law probability density distribution we present a review of applications highlight the common physical mechanism and summarize the main known results the distribution and statistical properties of the duration of intermittent bursts are also characterized in detail

a direct electrifying algorithm for backbone identification this paper proposes a new algorithm for identifying backbones in the application of percolation theory this algorithm is based on the current carrying definition of backbone and is carried out on the predetermined spanning cluster it is fairly easy to implement and further parallelize the efficiency is enhanced by the fact that the conductivity of a percolating system can be obtained in the same processing of backbone identification the critical exponents of backbone mass red bonds sites and conductivity obtained by this algorithm are in very good agreement with the existing results

a precise characterization of three dimensional percolating backbones the backbones of three dimensional critical percolation clusters are extracted and the fractal dimension is accurately calculated using various mass scaling and box counting techniques to be dfb 1 855 or 0 015 the eigenvalue spectrum of the brownian transition probability matrix is studied numerically which results in the estimates dwb 3 13 or 0 03 for the walk dimension and dsb 1 18 or 0 01 for the spectral dimension of the backbone independent calculation of all three quantities allows the conductivity exponent mu to be determined and also allows the validity of the scaling relation ds 2df dw proposed by alexander and orbach 1982 to be tested on the backbone

non linear conductivity of granular superconductors a novel breakdown problem the authors study a percolating network of highly hysteretic josephson junctions in which each junction is modelled as a piecewise ohmic hysteretic device at the percolation threshold the average voltage drop across a network on a large l l grid is assumed to fall to zero as v approximately i ic l eta as the applied current is reduced to the critical current ic l the finite size monte carlo study gives the value eta 2 0 or 0 1 this value is remarkably close to the result eta 2 obtained by a scaling analysis of a continuum version of the model

family of growth fractals with continuously tunable chemical dimension we introduce a new class of statistical growth fractals which is of interest because the chemical dimension d is continuously tunable we also study other exponents charac terising these fractals there has been considerable recent interest in uncovering the fashion in which the familiar laws of physics are modified for fractal objects in part because of the large number of important realisations of fractals in nature l 81 it has only recently become appreciated that the physics of fractals is determined by more than just the fractal dimension df which describes how the cluster mass n scales with the cluster radius r

shortest paths in percolation the separation of two points on a percolation network is characterised not only by the distance between them but also by the length of a path on the network which connects them the wetting velocity nu provides a measure of the lengths of the shortest connecting paths on the network above the percolation concentration pc along the easy axes nu is expected to vanish as p pc theta near pc while 1 nu is expected to vary as pd p theta near the directed percolation concentration pd the exponent theta is related to an exponent nu which characterises the shortest paths precisely at pc and which has recently been determined numerically the wetting velocity is calculated analytically on a randomly diluted bethe lattice and the values theta 1 2 and theta 1 with logarithmic corrections are found the direction dependence of nu is also investigated

static critical exponents from the dynamics of damage spreading and overhangs in the ising model with temperature gradient by studying damage spreading in spin models in a temperature gradient it is possible to determine not only the critical temperature and the correlation length exponent v but also a second critical exponent beta with high precision this is done by studying the dynamics of damage spreading in particular the overhangs which develop in the damage front the authors demonstrate the method for the two dimensional ising model using heat bath dynamics and discuss differences with metropolis dynamics

the data driven null models for information dissemination tree in social networks abstract for the purpose of detecting relatedness and co occurrence between users as well as the distribution features of nodes in spreading path of a social network this paper explores topological characteristics of information dissemination trees idt that can be employed indirectly to probe the information dissemination laws within social networks hence three different null models of idt are presented in this article including the statistical constrained 0 order idt null model the random rewire broken edge 0 order idt null model and the random rewire broken edge 2 order idt null model these null models firstly generate the corresponding randomized copy of an actual idt then the extended significance profile which is developed by adding the cascade ratio of information dissemination path is exploited not only to evaluate degree correlation of two nodes associated with an edge but also to assess the cascade ratio of different length of information dissemination paths the experimental correspondences of the empirical analysis for several sinaweibo idts and twitter idts indicate that the idt null models presented in this paper perform well in terms of degree correlation of nodes and dissemination path cascade ratio which can be better to reveal the features of information dissemination and to fit the situation of real social networks

decoupling of square singular systems via proportional state feedback necessary and sufficient conditions for the decoupling of a solvable square singular system ex t ax t bu t with output y t dx t through an admissible control law of the form u t kx t hr t where h is a square nonsingular matrix it has been shown that for a given singular system that satisfies these conditions a propagational state feedback exists for which the system s transfer function is a diagonal nonsingular and proper rational matrix the proofs of the main results are constructive and provide a procedure for computing an appropriate proportional state feedback

slimming of power law tails by increasing market returns abstract we introduce a simple generalization of rational bubble models which removes the fundamental problem discovered by lux and sornette j money credit and banking preprint at http xxx lanl gov abs cond mat 9910141 that the distribution of returns is a power law with exponent r when r is larger than the discount rate r the distribution of returns of the observable price sum of the bubble component and of the fundamental price exhibits an intermediate tail with an exponent which can be larger than 1 this regime r r corresponds to a generalization of the rational bubble model in which the fundamental price is no more given by the discounted value of future dividends we explain how this is possible our model predicts that the higher is the market remuneration r above the discount rate the larger is the power law exponent and thus the thinner is the tail of the distribution of price returns

nonlinear resistor fractal networks topological distances singly connected bonds and fluctuations the authors consider a fractal network of nonlinear resistors with the voltage v behaving as a power of the current i mod v mod r mod i mod alpha the resistance between two points at a distance l is r l varies as lzeta alpha they prove that zeta 0 describes the scaling of the topological chemical distance while zeta infinity describes that of the number of singly connected red bonds for random resistors they also consider the width of the resistance distribution delta r varies as l zeta 2 alpha values for zeta and zeta 2 are explicitly derived for two model fractals and delta r r is found to grow with l for the sierpinski gasket and alpha 1 612 the relevance of the results to percolation clusters is discussed

series expansion analysis of the backbone properties of two dimensional percolation clusters low density series expansions for the backbone properties of two dimensional bond percolation clusters are derived and analysed expansions for most of the 14 properties considered are new and are obtained to order on the square lattice and order on the triangular lattice earlier series work was confined to three properties of the square lattice and was to order the fractal dimension of the bonds or sites in the backbone is estimated to be and is intermediate between a previously conjectured field theory value and the latest monte carlo results the union intersection and length of the longest self avoiding paths are found to have the same fractal dimension which is close to and consistent with the field theory conjecture for on the other hand the union intersection and length of the shortest paths are found to have different dimensions and in the case of the intersection the triangular and square lattices are found to have sigificantly different dimensions the fractal dimension of the shortest path also appears to be non universal and we find for the square lattice and for the triangular lattice critical amplitude ratios are considered and found to be in agreement with theoretical inequalities

possibly exact fractal dimensions from conformal invariance based on an equivalence between the q state potts model and conformally invariant field theories the author speculates that all fractal dimensionalities of two dimensional percolation are given by the formula d 100 x2 48 where x is an integer the same formula should also apply to self avoiding walks and lattice animals this approach is extended to cluster weighted percolation his conjecture seems to agree with previous exact and numerical results except for the backbone exponent and possibly for lattice animals

scaling relation for the critical exponents of the backbone of percolation clusters the author suggests that beta b the critical exponent of the backbone of percolation clusters is given by beta b 1 2 gamma 5 beta 1 where gamma and beta are susceptibility and order parameter exponents of percolation respectively the proposed relation agrees with the epsilon expansion of beta b given by harris and lubensky and provides accurate estimates of beta b at all d 1 this relation may suggest that the backbone exponents are not independent of other percolation exponents the author also suggests a more general relation for the order parameter exponent of m connected clusters

effect of loops on the vibrational spectrum of percolation network abstract we study the effects of adding loops to a critical percolation cluster on the diffusional and equivalently scalar elastic properties of the fractal network from the numerical calculations of the eigenspectrum of the transition probability matrix we find that the spectral dimension ds and the walk dimension dw change suddenly as soon as the floppy ends of a critical percolation cluster are connected together to form relatively large loops and we find that the additional inclusion of successively smaller loops only change these exponents little if at all this suggests that there is a new universality class associated with the loop enhanced percolation problem

the decoupling of generalized state space systems via state feedback the input output decoupling problem of generalized state space systems gss via proportional state feedback is studied the following three major issues are resolved the necessary and sufficient conditions for the problem to have a solution the general analytical expressions of the controller matrices and the general analytical expressions of the diagonal elements of the closed loop system it is concluded that the approach presented seems to be a quite powerful tool for solving several feedback design problems for gss systems using this method problems such as decoupling and simultaneous pole assignment and decoupling via output feedback are successfully studied

distribution of dangling ends on the incipient percolation cluster we study numerically and by scaling arguments the probabilityp m dm that a given dangling end of the incipient percolation cluster has a mass between m and m dm we nd by scaling arguments that p m decays with a power law p m m 1 with an exponent d b f df where df and d b f are the fractal dimensions of the cluster and its backbone respectively our numerical results yield 0 83 in d 2 and 0 74 in d 3 in very good agreement with theory c 1999 published by elsevier science b v all rights reserved

regular random fractals and the n parameter model two models of fractal objects are presented the first model represents a new class of fractals which is intermediate to statistical fractals and exact fractals in that each configuration of the random ensemble has an exact and adjustable fractal dimension the second model has a continuously adjustable backbone dimensions and can be used to directly generate conducting backbones

geometrical properties of single connected bonds in percolation clusters numerical simulation of the two dimensional bond percolation at the percolation threshold on an l l lattice shows that the radius of gyration of singly connected bonds scb scales as l1 this result confirms an assumption used to derive a lower bound on the critical exponent of elasticity a persistent anisotropy in the distribution of scb is found and used to estimate the ratio between the bulk and shear moduli of a random percolating system local anisotropy in the direction of scb disappears for large l

topological properties of diffusion limited aggregation and cluster cluster aggregation the detailed topological or connectivity properties of the clusters formed in diffusion limited aggregation dla and cluster cluster aggregation cca are considered for spatial dimensions d 2 3 and 4 specifically for both aggregation phenomena the authors calculate the fractal dimension dmin nu 1 defined by l approximately rd min where l is the shortest path between two points separated by a pythagorean distance r for cca they find that dmin increases monotonically with d presumably tending toward a limiting value dmin 2 at the upper critical dimensionality dc as found previously for lattice animals and percolation for dla on the other hand they find that dmin 1 within the accuracy of the calculations for d 2 3 and 4 suggesting the absence of an upper critical dimension they also discuss some of the subtle features encountered in calculating dmin for dla

internal stabilization and decoupling in linear multivariable systems by unity output feedback compensation we consider the problem of internally stabilizing and simultaneously diagonally decoupling a linear multivariable system by unity output feedback compensation a sufficient condition is derived for the existence of a cascade proper compensator c s such that when employed in a unity feedback loop involving the proper transfer function matrix p o of a free of unstable hidden modes system sigma p o will not only internally stabilize the feedback closed loop system sigma p o c but will also give rise to a closed loop transfer function matrix h yr diag which is nonsingular diagonal and has desired poles based on this analysis an algorithmic procedure for the computation of such a compensator is presented

surface and edge exponents for the spreading of 3d percolation monte carlo results are presented for the spreading of 3d percolation where the seed consists of an infinite straight line this line can either be in the bulk of the material into which spreading is possible on a planar surface of the material or on a rectangular edge in the last two cases spreading occurs only into angular regions with 180 degrees resp 90 degrees while the mean distance grows at p pc in all three cases with the same exponent the average number of growth sites grows resp vanishes in all three cases with different exponents in particular this implies that at p pc the sites on the surface of a big cube belonging to the maximal cluster within that cube have fractal dimension 1

cluster structure near the percolation threshold derives exact relations that allow one to describe unambiguously and quantitatively the structure of clusters near the percolation threshold pc in particular the author proves the relations p dpij dp lambda ij where p is the bond density pij is the pair connectedness function and lambda ij is the average number of cutting bonds between i and j from this relation it follows that the average number of cutting bonds between two points separated by a distance of the order of the connectedness length xi diverges as mod p pc mod 1 the remaining multiply connected bonds in the percolating backbone which lump together in blobs diverge with a dimensionality dependent exponent he also shows that in the cell renormalisation group of reynolds et al 1978 1980 the thermal eigenvalue is simply related to the average number of cutting bonds in the spanning cluster he discusses a percolation model in which the blobs can be controlled by varying a parameter and study the influence on the critical exponents

fractal properties of the percolating backbone in three dimensions a recursive algorithm to determine the backbone of the two dimensional percolating cluster is adjusted to measure the fractal properties of the backbone of the three dimensional percolating cluster the specific properties measured are the fractal dimensionality of the backbone dbbf 1 75 or 0 04 the chemical dimensionality dt 1 26 or 0 03 and the minimum path dimensionality dmin 1 39 or 0 03

rational expectations in dynamic linear models analysis of the solutions in this paper we analyze the solutions of linear econometric models with rational expectations more precisely we describe in detail the set of all the solutions in particular this set is shown to be much larger than the sets previously considered we also study various criteria of selection in this set of solutions and we examine to what extent these criteria redtiuce the set of the solutions

input output decoupling of linear time varying singular systems the problem of input output decoupling by state feedback is considered for linear time varying singular systems assumptions are introduced and an algorithm is developed such that the system can be expressed in a simple normal form a time varying feedback law is then constructed which ensures under appropriate conditions that the closed loop system is regular impulse free and noninteractive

order propagation near the percolation threshold a new monte carlo method for studying bond percolation clusters is developed and used to identify new critical quantities associated with the percolation threshold the bonds in each cluster are partitioned into three distinct connectivity classes red singly connected backbone bonds blue multiply connected backbone bonds and yellow non backbone bonds often called dangling ends among the new cluster properties studied are the mean number of red bonds a critical quantity diverging at p c with exponent y r approximately 1 and the length of the shortest connected path through the cluster which is critical with exponent y min 1 35 0 02 for all cluster properties studies the authors also compute averages over only the largest clusters the corresponding critical exponents are found to be significantly different from those obtained by averaging over clusters of all sizes

shortest path fractal dimension for percolation in two and three dimensions we carry out a high precision monte carlo study of the shortest path fractal dimension d mathrm min for percolation in two and three dimensions using the leath alexandrowicz method which grows a cluster from an active seed site a variety of quantities are sampled as a function of the chemical distance including the number of activated sites a measure of the radius and the survival probability by finite size scaling we determine d mathrm min 1 130 phantom rule 0 16em 0ex 77 2 and 1 3756 6 in two and three dimensions respectively the result in two dimensions rules out the recently conjectured value d mathrm min 217 192 deng et al phys rev e 81 020102 r 2010

a new algorithm to extract the backbone in a random resistor network the authors present an efficient algorithm to determine the backbone the current carrying part of a lattice in the random resistor network problem it reduces mainly to the determination of the cluster structure on two lattices its efficiency is due to the use of the dual lattice and therefore is specific to the two dimensional case

position space renormalisation group study of shortest paths in percolation clusters the average length of the shortest paths along the backbone of the incipient infinite cluster is expected to diverge near the percolation threshold pc as mod p pc mod nu s with nu s different from the correlation length exponent nu the author applies the position space renormalisation group psrg technique to obtain nu s both for square and cubic lattices and for a triangular lattice in two dimensions the results obtained nu s 1 49 for d 2 and 1 21 for d 3 are in very good agreement with the existing results obtained by different methods

site and bond directed branched polymers for arbitrary dimensionality evidence supporting a relation with the lee yang edge singularity parisi and sourlas 1981 have argued that the isotropic branched polymer problem in dimension d 1 is related to the lee yang edge singularity problem in dimension d 1 to test if there is a relation with the directed branched polymer problem the authors calculate the generating functions for both site and bond directed branched polymers for arbitrary dimension d to order smax 10 and bmax 8 their analysis lends support to the proposal that theta d 1 1 theta d d sigma ly d 1 1 where theta and theta d are the critical exponents for lattice animals and directed lattice animals and sigma ly is the lee yang edge singularity exponent they also obtain expansions for the growth parameter in the variable sigma 1 2d 1 1 for bond and site directed lattice animals

universal substructures of percolation clusters the skeleton the authors define the skeleton of a cluster aggregate as the set of all sites belonging to the shortest paths connecting a chosen site with the lth chemical shell surrounding that site the fractal properties of skeletons of percolation clusters at criticality have been studied and it is inferred that the mass of the skeleton ms scales with the chemical distance l for l l as ms approximately lls where dls 1 is universal for l or d or 6 numerical evidence which supports this conclusion is presented for d 2 and an analytical proof is given for d 6

damage spreading in a gradient we propose a new method of analyzing the frozen chaotic transition in a cellular automaton by propagating damage in a gradient we obtain estimations forpc and for the critical exponents for the kauffman model and the mixture of or and xor rules

field theoretic approaches to biconnectedness in percolating systems two field theoretic formulations for the percolation problem are presented from which the critical exponents describing the backbone of the infinite cluster at the percolation threshold are obtained at high spatial dimension d the order parameter exponent for the backbone beta 2 is given by beta 2 2 beta psi 2 nu where beta is the critical exponent for the density of the infinite cluster and psi 2 is a new crossover exponent in mean field theory psi 2 0 and for d 6 epsilon psi 2 2 epsilon 2 49 o epsilon 3 presumably psi 2 d is a smooth function of d for d d where numerical and theoretical work indicates that d is about 3 the result indicates that the fractal dimensionality of the backbone is given in terms of the percolation exponents as gamma nu psi 2

the three dimensional ising model in a temperature gradient abstract the critical temperature t c and the correlation length exponent for the three dimensional ising model is determined via the gradient method by numerical studies of systems of size up to 360 3 we determine the correlation length exponent 0 62 0 01 in comparison to the best estimates ranging from 0 624 to 0 630

output feedback decoupling of generalized state space systems abstract the input output decoupling problem of generalized state space systems via pure proportional output feedback is studied here for the first time wherein the following four major issues are resolved the necessary and sufficient conditions for the problem to have a solution the general analytical expressions of the decoupling controller matrices the general analytical expressions of the diagonal elements of the decoupled closed loop system and the structural properties of the closed loop system