fractal measures of first passage time of a simple random walk abstract we consider random walks starting at the site i 1 on a one dimensional lattice segment with an absorbing boundary at i 0 and a reflecting boundary at i l we find that the typical value of first passage time fpt is independent of system size l while the mean value diverges linearly with l the qth moment of the fpt diverges with system size as l2q 1 for q 1 2 for a finite but large l the fpt distribution has an 1 t tail cut off by an exponential of the form exp t l2 however if l is set equal to infinity the distribution has an algebraic tail given by t 1 2 we find that the generalised dimensions d q have a nontrivial dependence on q this shows that the fpt distribution is a multifractal we also calculate the singularity spectrum f

fractals in physics squig clusters diffusions fractal measures and the unicity of fractal dimensionality the three topics discussed in this paper are largely independent part 1 fractal squig clusters are introduced and it is shown that their properties can match to a remarkable extent those of percolation clusters at criticality physics on these new geometric shapes should prove tractable as background the author s theories of squig intervals and squig trees are reviewed and restated in more versatile form part 2 the notion of latent fractal dimensionality is introduced and motivated by the desire to simplify the algebra of dimensionality scaling noises are touched upon a common formalism is presented for three forms of anomalous diffusion the ant in the fractal labyrinth fractional brownian motion and levy stable motion the fractal dimensionalities common to diverse shapes generated by diffusion are given in table i as functions of the latent dimensionalities of the support of the motion and of the diffusion itself part 3 it is argued that every fractal point set has a unique fractal dimensionality but it is pointed out that many fractals involve diverse combinations of many fractal point sets such is in particular the case for fractal measures and for fractal graphs often called hierarchical lattices the fractal measures that the author had introduced in the early 1970s are described including new developments

on the matrix difference i a for an nxn complex matrix a and the nxn identity matrix i n the difference i n ais investigated by exploiting a partitioned representation several features of such a difference are identified in particular expressions for its moore penrose inverse in some specific situations are established and representations of the pertinent projectors are derived special attention is paid to the problem how certain properties of a and i n a are related the properties in question deal with known classes of matrices such as gp ep partial isometries bi ep normal projectors and nilpotent an important part of the paper is devoted to demonstrating how to obtain representations of orthogonal projectors onto various subspaces determined by a and or i n a several such representations are provided and a number of relevant conclusions originating from them are identified

structure and perimeters of percolation clusters site percolation clusters are simulated at the percolation threshold on the square lattice an algorithm for walks around each cluster is used to obtain information on its fractal geometry the fractal dimensionality of the external perimeter is found to depend on the size of adsorbent particles used to measure it if the vacant perimeter sites have only nearest neighbour connectivity then the perimeter has dimension de 1 37 or 0 03 instead of that of the hull dh 1 75 the authors also measured the mass of internal dangling sites surrounded by the backbone s blobs singly and doubly connected sites the number of blobs and the average linear distance between entry into the exit from a blob

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

a regular model for cluster numbers and structure just below percolation threshold a two dimensional deterministic fractal model is proposed to imitate the geometric texture just below percolation threshold the model is constructed for bond percolation via a rule of bond occupation on the square lattice it show the typical percolation behaviour as a function of a parameter p p is the bond concentration the model describes the approach towards the threshold below criticality the critical bond concentration pc and the correlation length exponent nu are found the scaling property of the cluster size distribution is studied below the threshold the critical concentration and critical exponents are shown to agree with those derived from the regular model proposed for the geometric texture above the threshold the scaling relations for the critical exponents are shown to be satisfied

a regular model for bond percolation the approach towards the threshold a two dimensional regular model is constructed for bond percolation via a rule of bond deletions on the square lattice the model describes the approach towards the threshold the critical bond concentration pc and the correlation length exponent nu are found the fractal geometry of the infinite percolating network and the scaling property of the cluster size distribution are studied above the threshold the infinite percolating network is shown to be reconstructed by the mandelbrot koch curve proposed as the non random model for the infinite cluster at criticality

self dual fractal models for the low field hall effect near percolation threshold fractal lattices with the self duality property are proposed to imitate the geometric texture of two dimensional percolating networks at percolation threshold the exponents describing the power law dependence on the scale length of the hall and ohmic conductivities are found the exponent for the hall conductivity agrees with twice that for the ohmic conductivity the model is extended to describe the approach towards the threshold it is found that the model shows the typical percolation behaviour as a function of a parameter p the bond concentration

universality and superuniversality of multifractals in nonlinear resistor networks the multifractal functionf is generalized to describe noisy nonlinear random resistor networks an approximant function for the family of noise exponents is introduced that provides a good description of real percolative systems for strong nonlinearities by mapping from this family to the multifractal function one can approximate the latter a scale transformation of in the approximation makes the multifractal function universal for all nonlinearities and by applying an additional transformation this function becomes superuniversal i e independent of the dimension the universality is demonstrated for the mandelbrot given structure and the implications of these results are discussed on real percolative systems

analytical calculation of two leading exponents of the dilute potts model a potts model on a square lattice with two and four spin interaction and site and bond dilution is shown to be dual to itself the model is mapped onto a vertex problem which in turn is equivalent to a solid on solid model by means of these mappings the dilute potts model can be written as a gaussian like model with staggered and direct periodic fields the leading and next to leading exponents of the potts model are calculated subject to the validity of certain assumptions

constructing projections on sums and intersections abstract in this note we list several formulas for computing orthogonal projections onto the linear sum and intersection of two subspaces of c n in terms of the projections on the individual subspaces

multifractality and the kauffman model the author simulates the spread of an initial damage in the kauffman cellular automata at its critical point on the square lattice he calculates various moments of the probability that a site has been damaged n times and checks for multifractality in the critical exponents for different ensembles specifically he finds no evidence for multifractality when the moments of the probabilities are evaluated with lattice size l but multifractal behaviour occurs when the moments are monitored as a function of time t

topological properties of percolation clusters percolation properties including the total cluster mass s the shell mass b and the linear geometrical size r are studied as a function of the topological chemical distance parameter l all critical exponents are shown to be related to an apparently new exponent nu defined by r approximately lv critical exponents are calculated exactly for percolation clusters on the cayley tree a model for 6d percolation for which s approximately l2 b approximately l and r2 approximately l for diffusion on such clusters one finds that l3 approximately t numerical estimates of the exponents are obtained for other dimensions a conjecture which relates nu to beta and nu is discussed

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

wetting velocity near the directed percolation threshold near the directed percolation threshold pd the wetting velocity upsilon scales as 1 upsilon approximately pd p theta the authors extrapolate from calculations on strips of finite width to show that on the two dimensional square lattice theta 1 73 or 0 02 this agrees with a scaling argument by barma and ray phys rev b vol 34 p 3403 1986 which predicts theta nu sub

further results on generalized and hypergeneralized projectors abstract the notions of generalized and hypergeneralized projectors introduced by gros and trenkler j gro s j trenkler generalized and hypergeneralized projectors linear algebra appl 264 1997 463 474 are revisited on the one hand the present paper provides several new characterizations of these sets and on the other the properties of generalized and hypergeneralized projectors related to various matrix partial orderings are considered moreover the paper demonstrates the usefulness in studying the properties of generalized and hypergeneralized projectors of the representation of complex matrices given in corollary 6 by hartwig and spindelbock r e hartwig k spindelbock matrices for which a and a commute linear and multilinear algebra 14 1984 241 256

motion of curves in three spatial dimensions using a level set approach the level set method was originally designed for problems dealing with codimension one objects where it has been extremely succesful especially when topological changes in the interface i e merging and breaking occur attempts have been made to modify it to handle objects of higher codimension such as vortex filaments while preserving the merging and breaking property we present numerical simulations of a level set based method for moving curves in r3 the model problem for higher codimension that allows for topological changes a vector valued level set function is used with the zero level set representing the curve our results show that this method can handle many types of curves moving under all types of geometrically based flows while automatically enforcing merging and breaking

phase transitions on fractals ii sierpinski gaskets for pt i see ibid vol 16 p 1267 1983 the authors construct and investigate a family of fractals which are generalisations of the sierpinski gaskets sgs to all euclidean dimensionalities these fractal lattices have a finite order of ramification and can be considered marginal between one dimensional and higher dimensional geometries physical models defined on them are exactly solvable the authors argue that short range spin models on the sg show no finite temperature phase transitions as examples they solve a few spin models and study the resistor network and percolation problems on these lattices

are there infinitely many independent exponents in the percolation cluster numbers abstract an infinite number of independent exponents is expected not in the average cluster numbers but in the tails of the their probability distribution this effect is somewhat similar to voltage distributions in random resistor networks

critical dynamics in heisenberg ferromagnets and antiferromagnets near the percolation threshold the critical spin wave dynamics of dilute ferromagnetic and antiferromagnetic heisenberg systems near the percolation threshold is studied on a two dimensional regular fractal model for bond percolation which captures the relevant geometric features of the cluster structure the dynamic critical exponent and spectral dimension for both the ferromagnetic and antiferromagnetic systems are calculated the treatment of the dynamics in these systems requires the use of generalised scaling techniques for sublattice systems which involve an extension of the parameter space

low concentration series in general dimension we discuss recent work on the development and analysis of low concentration series for many models the recent breakthrough in the extremely efficient no free end method of series generation facilitates the derivation of 15th order series for multiple moments in general dimension the 15th order series have been obtained for lattice animals percolation and the edwards anderson ising spin glass in the latter cases multiple moments have been found from complete graph tables through to 13th order general dimension 13th order series have been derived for the resistive susceptibility the moments of the logarithms of the distribution of currents in resistor networks and the average transmission coefficient in the quantum percolation problem 11th order series have been found for several other systems including the crossover from animals to percolation the full resistance distribution nonlinear resistive susceptibility and current distribution in dilute resistor networks diffusion on percolation clusters the dilute ising model dilute antiferromagnet in a field and random field ising model and self avoiding walks on percolation clusters series for the dilute spin 1 2 quantum heisenberg ferromagnet are in the process of development analysis of these series gives estimates for critical thresholds amplitude ratios and critical exponents for all dimensions where comparisons are possible our series results are in good agreement with bothe expansion results near the upper critical dimension and with exact results when available in low dimensions and are competitive with other numerical approaches in intermediate realistic dimensions

a class of codimension two free boundary problems this review collates a wide variety of free boundary problems which are characterized by the uniform proximity of the free boundary to a prescribed surface such situations can often be approximated by mixed boundary value problems in which the boundary data switches across a codimension two free boundary namely the edge of the region obtained by projecting the free boundary normally onto the prescribed surface as in the parent problem the codimension two free boundary needs to be determined as well as the solution of the relevant field equations but no systematic methodology has yet been proposed for nonlinear problems of this type after presenting some examples to illustrate the surprising behavior that can sometimes occur we discuss the relevance of traditional ideas from the theories of moving boundary problems singular integral equations variational inequalities and stability finally we point out the ways in which further refinement of these techniques is needed if a coherent theory is to emerge

a family of crossover exponents for the nonlinear resistor network the crossover exponents phi 2 r and phi 3 r for the nonlinear resistor network are calculated to first order in in 6 d using the same formalism used by harris 1987 the result supports the idea of the analytic continuation proposed by harris in calculating the nonlinear crossover exponent phi r for the nonlinear resistor network

a regular random fractal model for cluster numbers and structure in percolation a regular random fractal intermediate between statistical fractals and deterministic fractals is presented to imitate the geometric texture at and near percolation threshold the model is constructed on the square lattice via a rule of bond occupation with use of position space renormalisation group it shows the typical percolation behaviour as a function of a parameter p the bond concentration the critical bond concentration correlation length exponent and scaling property of the cluster size distribution are found the scaling relations between critical exponents for cluster numbers and structure are shown to be exactly satisfied the fractal dimension of its cutting bonds agrees with the inverse of the connectedness length exponent at criticality

thermodynamical formalism for an infinite hierarchy of fractal resistor networks where n is the number of resistors we find a generic phase transition at 3 3c 1 also we develop a transfer matrix technique which determines all even positive moments the thermodynamical formalism is applied to the hilfer blumen hierarchy of generalized sierpinski gasket fractal networks and the crossover from fractal to lattice behavior is studied at this crossover we find a sharp phase transition in the second moment 3 2

some general relations between dressed self avoiding walks and percolation perimeters on lattices the author shows for some simple examples the relations between some recently proposed dressed self avoiding walks and site percolation connection between matching pairs and dressed walk generators is specified and some features of the corresponding bond percolation problem are given the case of surface generators for dressed self avoiding surfaces is briefly examined

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

fractal models for diffusion controlled aggregation a two dimensional fractal model is constructed for diffusion controlled deposition on a surface the fractal geometry of the deposit and the power law behaviour of the quantities characterising the non equilibrium cluster size distribution are shown to be consequences of the competition generally present in a nonlinear growth process a qualitative agreement with previous numerical results is found and the scaling laws for the critical exponents of the problem are shown to be satisfied exactly

the moore penrose inverse of a partitioned nonnegative definite matrix abstract consider an arbitrary symmetric nonnegative definite matrix a and its moore penrose inverse a partitioned respectively as a e f f h and a g 1 g 2 g 2 g 4 explicit expressions for g 1 g 2 and g 4 in terms of e f and h are given moreover it is proved that the generalized schur complement a g 4 g 1 g 2 g 4 g 2 is always below the moore penrose inverse a h of the generalized schur complement a h e fh f with respect to the lowner partial ordering

probability densities of homogeneous functions explicit approximation and applications to percolating networks the author treats the probability density of homogeneous functions of independent variables assuming these variables are narrowly distributed he finds an explicit approximate form for the laplace transform of the function s density this expression is used to show that if the variables are all gaussian then so is the global function the latter result is also claimed to hold for the total resistance of any random resistor network between two arbitrary terminals he further utilises the result to explicitly find the laplace transform of the conductance density of i a hierarchical structure and ii a random network in 1 in dimensions using the migdal kadanoff procedure in the last case he also finds the scaling of all the cumulants and finds that they exhibit a constant gap

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

a hierarchical model for scaling structure in generalised diffusion limited aggregations deterministic fractal models are presented to be exactly solvable for the growth probability distribution on the surface of the cluster in a hierarchical lattice the recursion relations of the electric field on the growth bond are obtained for diffusion limited aggregation and the dielectric breakdown models a hierarchy of generalised dimensions d q is calculated to describe the growth probability by using the recursion relations the partition of q 1 d q into a density of singularities f q with singularity strength alpha q is made and the alpha f spectra are studied for different dielectric breakdown models the scaling of the highest growth probability pmax on the growth bond is analytically derived and the dependence of the fractal dimensions is found on the parameter eta describing the different dielectric breakdown models

a fractal model for band structures just above percolation threshold a two dimensional regular fractal model is proposed to imitate the geometric texture of percolating networks just above percolation threshold the system is self similar fractal on length scales smaller than the connectedness length but becomes homogeneous non fractal square lattice on larger length scales the scaling of vibrational density of states and band structures are studied by using exact renormalisation group methods it is shown that an infinite number of narrow bands appear to have self similar structures of typical cantor sets at low frequency

exact fractals with adjustable fractal and fracton dimensionalities a family of exact fractals is presented by adjusting two external parameters a wide range of fractal and fracton dimensionalities can be achieved this includes the case of fracton dimensionality of 2 which is critical for diffusion

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

squig sheets and some other squig fractal constructions squig intervals are a class of hierarchically constructed fractals introduced by the author they can be visualized as the final outcome upon a straight interval of a suitable cascade of local perturbative eddies ruled by two processes called decimation and separation their theory is summarized and their scope is extended in several new directions especially by introducing new forms of separation squig intervals are generalized in two dimensions with fractal dimensions ranging from 1 2886 to 1 589 squig sheets are constructed in three dimensional space with fractal dimensions ranging from 8 3 up they should prove useful in modeling the fractal surfaces associated with turbulence and related phenomena squig intervals are constructed in three dimensions nonsymmetric eddies and the resulting squigs are tackled squig trees and intervals are drawn on unconventional lattices either in the plane or in a prescribed fractal surface peyriere sm systems are mentioned their study includes the proof that the informal renormalization argument involving a transfer matrix is exact for squigs

on the spreading dimension of percolation and directed percolation clusters the authors study the number an of sites that are accessible after n steps at most on clusters at the percolation threshold on a cayley tree an is of order n2 if the origin belongs to a larger cluster whereas its average over all clusters is of order n this suggests that the intrinsic spreading dimension d defined by an approximately nd is equal to two for fractal percolation clusters in space dimensions d or 6 and depends on d for d 6 for directed percolation clusters they argue that d is related to usual critical exponents by d beta gamma nu sub monte carlo data that support this relation are presented in two dimensions analogous results are derived for lattice animals d 2 on the cayley tree and d 1 nu sub for directed animals in any dimensions

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

renormalisation group approach to multifractal structure in random resistor networks just above the percolation threshold a renormalisation group method is presented to analyse the multifractal structure of the current distribution in random linear resistor networks just above the percolation threshold the recursion relation for the current distribution is given under the renormalisation transformation the distribution function of the current is derived with the use of the recursion relation the fragmentation of the current is described as a random multiplicative process an infinite set of exponents is calculated to scale each of the moments of the current distribution the alpha f spectrum is derived from the legendre transform of these exponents the authors exponents obtained by the relatively small cell renormalisation are very close to the previous simulation data

search for multifractality in damage spreading for kauffman cellular automata the authors check how an initial disturbance called the damage spreads through a square lattice of kauffman cellular automata at their critical point they determine the moments of the probability that a site has been damaged n times and check for multifractality in the fractal dimensions or critical exponents of these moments of the damage probability

on a matrix decomposition of hartwig and spindelbock abstract the main task of the paper is to demonstrate that corollary 6 in r e hartwig k spindelbock matrices for which a and a commute linear and multilinear algebra 14 1984 241 256 provides a powerful tool to investigate square matrices with complex entries this aim is achieved on the one hand by obtaining several original results involving square matrices and on the other hand by reestablishing some of the facts already known in the literature often in extended and or generalized forms the particular attention is paid to the usefulness of the aforementioned corollary to characterize various classes of matrices and to explore matrix partial orderings

on the projectors ff and f f abstract a particular version of the singular value decomposition is exploited for an extensive analysis of two orthogonal projectors namely ff and f f determined by a complex square matrix f and its moore penrose inverse f various functions of the projectors are considered from the point of view of their nonsingularity idempotency nilpotency or their relation to the known classes of matrices such as ep bi ep gp dr or sr this part of the paper was inspired by benitez and rakocevic j benitez v rakocevic matrices a such that aa a a are nonsingular appl math comput 217 2010 3493 3503 further characteristics of ff and f f with a particular attention paid on the results dealing with column and null spaces of the functions and their eigenvalues are derived as well besides establishing selected exemplary results dealing with ff and f f the paper develops a general approach whose applicability extends far beyond the characteristics provided therein

series expansion study of the distribution of currents in the elements of a random diode insulator network the critical exponent zeta k of the kth moment of the current distribution for random diode insulator networks on the square and simple cubic lattices is calculated for a range of k using low density series expansion techniques it is also shown that zeta 1 nu sub where nu sub is the critical exponent of the parallel connectedness length for directed percolation the values of zeta k for the simple cubic lattice are well fitted by a simple exponential formula with zeta k 1 zeta k 1 1 1 2 the skal shklovskii scaling relation for the conductivity is generalised to the kth moment and it follows that the exponent k describing the divergence of flicker noise is given by kappa approximately d 1 nu perpendicular to nu sub zeta 4 2 zeta 2 the authors estimate this for both lattices

meshless projection iterative analysis of signorini problems using a boundary element free method this paper presents a meshless method based on coupling a boundary element free method befm with a new projection iterative scheme for signorini problems using the new iterative scheme the nonlinear complementary signorini boundary conditions are transformed into neumann boundary conditions then mixed elliptic problems are formed and are solved by the befm in this method only a nodal data structure on the boundary is required and the coefficient matrices of the discrete system equations only have to be computed once in the whole iterative process convergence of the iterative scheme is discussed theoretically numerical examples involving beach percolation and an electropaint process are given to illustrate the convergence rate and high computational efficiency of the method

multifractal structure of clusters and growing aggregates abstract various phenomena on a given fractal object have different critical behavior this is related to the underlying multifractal structure of the aggregate here we discuss the problem with particular emphasis on a scaling approach which leads to a simple real space renormalization group a mechanism which generates multifractality based on a multiplicative process is also illustrated the multifractal structure of the incipient infinite cluster in percolation and diffusion limited aggregation is discussed in detail

diffusion on fractal lattices and the fractal einstein relation finitely ramified fractal lattices show anomalous diffusion with r2 varies as t2h there is a hierarchy of transit times which are shown by monte carlo simulation to satisfy tau n varies as alpha n for large n where alpha b1h and b is the base of the lattice the lattice resistivity scales with size as rho varies as rn where r is characteristic of the lattice exact renormalisation group methods yield alpha and r analytically and verify the einstein relation for fractals alpha rn