analytic solutions of the fisher equation the authors analyse the fisher equation via the expanded painleve analysis approach they obtain its singular property auto backlund transformation and analytic solutions including some interesting heteroclinic and homoclinic solutions

invariant solutions of certain nonlinear evolution type equations with small parameters the fisher equation which arises in the study of reaction diffusion waves in biology does not display a high level of symmetry properties consequently only travelling wave solutions are obtainable using the method of invariants this has a direct bearing on studying perturbed forms of the equation which may arise from considering e g damping or dissipative factors we show here how one can get around this limitation by appending some unknown function to the perturbation and obtain interesting practical results using invariants the ideas have significant consequences for equations which do not admit large class of symmetry properties the method used in this analysis is then extended to other classes of evolution type equations that involve perturbations for e g the kdv type equations

a sufficient condition for stability of a polytope of matrices abstract in this paper we study the stability of interval dynamical systems a sufficient condition for the stability of a polytope of matrices which is shown to be necessary and sufficient for a certain class of matrices is obtained the results developed can also be applied to the stability of a positive cone of matrices and sufficient conditions for the stability of interval dynamical systems are obtained a relationship between real parts of eigenvalues and matrix measures is also presented

stability analysis of gray discrete time systems the stability of discrete time system whose state matrix is an interval matrix is discussed in terms of lyapunov stability theory some sufficient conditions are obtained which guarantee the stability of gray discrete time systems or interval discrete time systems illustrative examples are given

improved stability robustness bounds using state transformation for linear discrete systems abstract sufficient condition bounds on the linear time varying perturbations for robust stability of an asymptotically stable linear time invariant discrete time system have been reported recently the development involves both schwartz and triangular inequalities and tends to give conservative results this paper uses state transformation to obtain improvement in these bounds both unstructured and structured perturbation bounds are shown to improve using transformations the proposed analysis is applied to a fourth order macroeconomic system model

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

nonlinear evolution type equations and their exact solutions using inverse variational methods we present the role of invariants in obtaining exact solutions of differential equations firstly conserved vectors of a partial differential equation p d e allow us to obtain reduced forms of the p d e for which some of the lie point symmetries in vector field form are easily concluded and therefore provide a mechanism for further reduction secondly invariants of reduced forms of a p d e are obtainable from a variational principle even though the p d e itself does not admit a lagrangian in this latter case the reductions carry all the usual advantages regarding noether symmetries and double reductions the examples we consider are nonlinear evolution type equations such as the korteweg devries equation but a detailed analysis is made on the fisher equation which describes reaction diffusion waves in biology inter alia other diffusion type equations lend themselves well to the method we describe e g the fitzhugh nagumo equation which is briefly discussed some aspects of painleve properties are also suggested

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

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 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

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

comments on sufficient and necessary condition for the asymptotic stability of discrete linear interval systems abstract in a recent paper by jiang 1988 a necessary and sufficient condition for asymptotic stability of the interval matrix of discrete time systems is presented in this contribution it is shown that this condition is not true in general a counter example is given and additional remarks are made

sufficient conditions on stability of interval matrices connections and new results the stability properties of interval matrices are discussed sufficient conditions are given for these properties based upon gershgorin s theorem and its extension and they are shown to improve previous results the derivation helps clarify relations between certain previously available results and shows they can be derived using gershgorin s theorem furthermore the problem of determining the stability of interval matrices is related to that of characterizing certain nonsingular m matrices

a new numerical scheme for the fisher equation in the context of the one dimensional fisher equation 1937 the authors study a new numerical scheme for reaction diffusion equations proposed by oono and puri 1987 they find that this scheme enables a reasonable simulation of the fisher equation at high values of the time increment where conventional schemes are not applicable for lower values of the time increment the new scheme compares favourably with conventional schemes

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

conservation laws of high order nonlinear pdes and the variational conservation laws in the class with mixed derivatives the construction of conserved vectors using noether s theorem via a knowledge of a lagrangian or via the recently developed concept of partial lagrangians is well known the formulas to determine these for higher order flows are somewhat cumbersome but peculiar and become more so as the order increases we carry out these for a class of high order partial differential equations from mathematical physics and then consider some specific ones with mixed derivatives in the latter set of examples our main focus is that the resultant conserved flows display some previously unknown interesting divergence properties owing to the presence of the mixed derivatives overall we consider a large class of equations of interest and construct some new conservation laws

counter example to sufficient and necessary condition for the asymptotic stability of discrete linear interval systems abstract in this note the interval matrix result for discrete linear interval system by jiang 1988 is shown to be incorrect i e all eigenvalues of each matrix a e a 1 an interval matrix have magnitudes less than 1 if and only if all the eigenvalues of every extreme matrix of the interval matrix a 1 have magnitudes less than 1

the distribution of the reflection phase of disordered conductors the reflection coefficients of one dimensional disordered conductors is asymptotically a pure phase when the sample length is much larger than the localisation length the authors study the stationary distribution of this reflection phase for tight binding chains with diagonal disorder i e random site potentials they derive the dyson integral equation obeyed by this distribution and relate it to the lyapunov exponent and the integrated density of states this general approach allows them to consider then more specifically two types of disorder in the case of a binary alloy where the potentials take two values the distribution of the reflection phase is shown to have generically an infinity of power law singularities related to the occurrence of periodic patterns of potentials this phenomenon produces divergences in the distribution when the strength of disorder exceeds some energy dependent threshold they also obtain an exact expression for the distribution of the reflection phase for a symmetric exponential distribution of potentials this non trivial solvable model allows them to examine analytically various properties of the phase distribution such as the anomalies which occur at weak disorder

brief paper simultaneous schur stability of interval matrices interval matrix structures are ubiquitous in nature and engineering ordinarily in an uncertain system there is associated with a set of coupled interval matrices a basic issue of exploring its asymptotic stability here we introduce the notion of simultaneous schur stability by linking the concepts of the majorant and the joint spectral radius and prove the asymptotic stability of a set of interval matrices governed by simultaneous schur stability the present result may lead to the stability analysis of discrete dynamical interval systems

counter examples to sufficient and necessary condition for the asymptotic stability of discrete linear interval systems jiang 1988 gives a sufficient and necessary condition for the asymptotic stability of a discrete linear interval system it is here demonstrated through counter examples that the condition is not sufficient the flaw leading to this erroneous result is also indicated

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

sufficient condition for asymptotic stability of discrete interval systems abstract a counter example is given to a recent result on the stability analysis of interval matrices by jiang 1988 a sufficient condition is then presented for the asymptotic stability of the discrete time interval system x k 1 a i x k by testing the norms of extreme matrices of the interval matrix a i such that they are all less than one

an inequality for the spectral radius of an interval matrix abstract for an n n interval matrix a aij we say that a is majorized by the point matrix a a ij if aij aij when the jth column of a has the property that there exists a power a m containing in the same jth column at least one interval not degenerated to a point interval and aij aij otherwise denoting the generalized spectral radius in the sense of daubechies and lagarias of a by a and the usual spectral radius of a by a it is proved that if a is majorized by a then a a this inequality sheds light on the asymptotic stability theory of discrete time linear interval systems

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

necessary and sufficient conditions for stability of a matrix polytope with normal vertex matrices abstract it is shown that the eigenvalue set of a matrix polytope with normal vertex matrices is a subset of the convex hull of eigenvalues of the vertex matrices

on the stability of discrete time linear interval systems abstract a general contraction mapping based method is proposed for studying the stability of linear interval systems sufficient conditions for the system uniform asymptotic stability using the products of vertex matrices are provided

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

counter examples to jiang s results abstract this note points out an important mistake in a recent paper by jiang 1988 on the asymptotic stability of discrete linear interval systems

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

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

necessary and sufficient conditions for stability of symmetric interval matrices it is shown that a family of symmetric interval matrices is schur if and only if the vertex matrices are schur this result is also applicable to hurwitz stability and aperiodic properties of linear systems

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

comments on stability analysis of interval matrices improved bounds it is shown that the main result of the previous paper is correct only for interval matrices with linearly varying elements not for the general class of interval matrices

on possibilities of the extension of kharitonov s stability test for interval polynomials to the discrete time case it is shown that the kharitonov test for hurwitz stability of an interval polynomial does not extend in general to the discrete time case unless the degree n of the polynomial is not greater than two for n leq 3 a given monic interval polynomial has all roots inside the unit disk if all 2 3 8 extreme polynomials have that property instead of only four polynomials in kharitonov s test for n 4 it is shown by a counterexample that discrete time stability of all extreme polynomials does not guarantee the stability of the interval polynomial

travelling wave and similarity solutions to nonlinear evolution type equations through inverse variational and symmetry methods abstract shallow water equations are usually modelled by nonlinear kdv type equations of which various generalisations now exist for example there are vector versions of the modified kdv equation and shallow water equations with nonlinear internal waves we discuss the reduction and solutions of these and other large classes of such type of equations using inverse variational and symmetry methods

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

comments on sufficient and necessary condition for the asymptotic stability of discrete linear interval systems in a recent paper by jiang 1988 a necessary and sufficient condition is given for a set of real interval matrices to have all their eigenvalues in the unit circle this result is shown by counter example to be false and the flaw in the proof is pointed out

solitary wave solutions to some classes of nonlinear evolution type equations using inverse variational methods abstract invariants of reduced forms of a p d e are obtainable from a variational principle even though the p d e itself does not admit a lagrangian the reductions carry all the advantages regarding noether symmetries and double reductions via first integrals or conserved quantities the examples we consider are nonlinear evolution type equations like the general form of the fizhugh nagumo and kdv burgers equations some aspects of painleve properties of the reduced equations are also obtained

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

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

counter example to a recent result on the schur stability of interval matrices by c l jiang abstract jiang 1988 stated that the schur stability of an interval matrix a i is equivalent to that of the vertices of a i we point out via a counter example that this result is incorrect

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

convergence property of interval matrices and interval polynomials in association with robust control system design and analysis the hurwitz property of interval matrices and interval polynomials has recently been actively investigated however its discrete counterpart the convergence property has seemingly not been much discussed in this paper this property is studied in comparison with the hurwitz counterpart some conditions under which interval matrices or interval polynomials are convergent are derived

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

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

strongly disordered chain of impedances theoretical analysis and numerical results the authors study the input impedance of a chain of random resistors chosen according to a broad distribution i e decaying as a power law for large arguments as a function of frequency the modulus of the input impedance varies as a power law while its phase has a well defined value a simple theoretical analysis based on estimations of sums of random variables yields the value of the exponents which are in perfect agreement with numerical simulations even when logarithmic corrections come into play the analysis can be extended further to investigate the sample to sample fluctuations of the impedance again agreement with simulations is very good

comments on convergence property of interval matrices and interval polynomials this note shows that the discrete linear interval system result of mori and kokame 1987 is also applicable to linear discrete time systems with linearly dependent perturbations

sufficient and necessary condition for the asymptotic stability of discrete linear interval systems in this paper a sufficient and necessary condition for the asymptotic stability of the discrete linear interval system x k 1 a1x k is presented i e all the eigenvalues of each matrix a a1 an interval matrix have magnitudes less than 1 if and only if all the eigenvalues of every extreme matrix of the interval matrix a1 have magnitudes less than 1