subgradients of optimal value functions in dynamic programming the case of convex systems without optimal paths we study the first order behaviour of the optimal value function associated to a convex dynamic programming problem the optimization process takes place in a certain environment characterized by some perturbation parameters affecting the transition costs and or the evolution law of the dynamic system an important aspect of this work is that we do not assume the existence of optimal paths to the unperturbed problem

introduction to matrix analytic methods in stochastic modeling preface part i quasi birth and death processes 1 examples part ii the method of phases 2 ph distributions 3 markovian point processes part iii the matrix geometric distribution 4 birth and death processes 5 processes under a taboo 6 homogeneous qbds 7 stability condition part iv algorithms 8 algorithms for the rate matrix 9 spectral analysis 10 finite qbds 11 first passage times part v beyond simple qbds 12 nonhomogeneous qbds 13 processes skip free in one direction 14 tree processes 15 product form networks 16 nondenumerable states bibliography index

continuum percolation in two dimensions monte carlo tests of scaling and universality for non interacting discs detailed results are reported for the connectivity properties of a system of discs of unit radius free to be situated anywhere within a square of area 2l2 ordinary lattke percolation would correspond to the discs being situated on the vertices of a j2l x j2l lattice computer simulations are carried out for a sequence of increasing system sizes ranging from l 20 to l 1000 for each value of l a large number of realisations are generated for 25 values of the disc concentration x we calculate a variety of estimates for the threshold parameter x as well as the critical exponents p y t and v our exponent estimates are in close agreement with accepted values for ordinary lattice percolation therefore this continuum system appears to be in the same universality class as lattice percolation

an s s random lifetime inventory model with a positive lead time positive lead times substantially complicate the modeling and analysis of inventory systems with finite shelf lifetimes and they have not been sufficiently addressed in the existing literature in this paper we analyze an s s continuous review model with a positive lead time we assume an exponential lifetime and an exponential lead time matrix geometric solutions can be obtained for the steady state probability distribution of the inventory level we then derive the total expected cost function we carry out numerical studies and gain insights to the selection of system parameters the findings on the impact of a positive lead time on the optimal reorder point and reorder batch size will be useful in developing strategies in handling inventory problems with finite lifetimes and positive lead times

necessary conditions for optimal control problems with state constraints necessary conditions of optimality are derived for optimal control problems with pathwise state constraints in which the dynamic constraint is modelled as a differential inclusion the novel feature of the conditions is the unrestrictive nature of the hypotheses under which these conditions are shown to be valid an euler lagrange type condition is obtained for problems where the multifunction associated with the dynamic constraint has values possibly unbounded nonconvex sets and satisfies a mild one sided lipschitz continuity hypothesis we recover as a special case the sharpest available necessary conditions for state constraint free problems proved in a recent paper by ioffe for problems where the multifunction is convex valued it is shown that the necessary conditions are still valid when the one sided lipschitz hypothesis is replaced by a milder local hypothesis a recent dualization theorem permits us to infer a strengthened form of the hamiltonian inclusion from the euler lagrange condition the necessary conditions for state constrained problems with convex valued multifunctions are derived under hypotheses on the dynamics which are significantly weaker than those invoked by loewen and rockafellar to achieve related necessary conditions for state constrained problems and improve on available results in certain respects even when specialized to the state constraint free case proofs make use of recent decoupling ideas of the authors which reduce the optimization problem to one to which pontryagin s maximum principle is applicable and a refined penalization technique to deal with the dynamic constraint

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

stochastic analysis of a continuous review perishable inventory system with positive lead time and poisson demand abstract this contribution is a study of the stochastic process associated with the inventory level of a continuous review perishable inventory system of s s type the demands to the inventory system are governed by a homogeneous poisson process the model considered in the paper admits positive lead time the inventory is reviewed continuously the usable age of a batch of arriving items is a known constant explicit expression for the stationary distribution of the stochastic process representing the level inventory is derived for the lost sales case under a specified aging phenomena of a batch of items the stationary distribution of the inventory level process is used to obtain a closed form expression for the total expected cost per unit time in the stationary case in operating the s s policy which may be used as an objective to determine the optinal reorder level the cost expression is comprehensive and considers all the relevant costs like the fixed ordering cost proportional unit ordering cost cost of shortage and storage cost due to items that perish in a batch also certain first passage time distributions associated with the inventory level stochastic process which are of special interest in terms of the inventory system are derived several known special cases of our results are deduced a simple numerical example is provided

statistics of red sites on elastic and full backbone abstract we investigated the number of red sites on the elastic and real backbone when right at the percolation threshold a spanning cluster exists between two sites at opposite faces of the lattice and found that it scales in the same way as in the case of percolation between two plates we also found that the number of common red sites scale similarly for both kinds of backbones for percolation between pairs of sites on opposite faces of the lattice our statistics for several quantities show that the exponent for the elastic backbone approaches the one for the full backbone as dimensionality is increased

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

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 discrete time model for common lifetime inventory systems we consider a discrete time s s inventory model in which the stored items have a random common lifetime with a discrete phase type distribution demands arrive in batches following a discrete phase type renewal process with zero lead time and allowing backlogs we construct a multidimensional markov chain to model the inventory level process we obtain a closed form expected cost function numerical results demonstrate some properties of optimal ordering policies and cost functions when compared with the results for the constant lifetime model the variance of the lifetime significantly affects the system behavior thus the formalism that we create here adds a new dimension to the research in perishable inventory control under uncertainty in lifetime

moment of inertia of doubly connecting bonds in two dimensional bond percolation by applying a theorem from conformal field theory to bond percolation in two dimensions the author obtains an equation for the second moment i of bonds which doubly connect a percolation cluster i a 5 square root 3 27 pi 2 p pc 2 the equation is exact as the occupancy probability p approaches the critical probability pc from either side a is a calculable lattice dependent number which for a square lattice equals 1 2

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

uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation for independent translation invariant irreducible percolation models it is proved that the infinite cluster when it exists must be unique the proof is based on the convexity or almost convexity and differentiability of the mean number of clusters per site which is the percolation analogue of the free energy the analysis applies to both site and bond models in arbitrary dimension including long range bond percolation in particular uniqueness is valid at the critical point of one dimensional 1 x y 2 models in spite of the discontinuity of the percolation density there corollaries of uniqueness and its proof are continuity of the connectivity functions and except possibly at the critical point of the percolation density related to differentiability of the free energy are inequalities which bound the specific heat critical exponent in terms of the mean cluster size exponent and the critical cluster size distribution exponent e g 1 2 1 1

on backbends on percolation backbones just above the percolation concentration a path on the backbone which leads from one side of the lattice to the other is not direct by zigzags through the lattice backbends are the portions of the zigzags which go backwards they are important in the problem of particle transport in strong external fields as they act as traps and limit the current the threshold concentration for the proliferation of paths with backbends no longer than a given length l is defined as pb l with the limits pb 0 pd directed percolation and pb infinity pc ordinary percolation the inverse function zeta p is the smallest integer such that for given p between pc and pd there are paths to infinity on which every backbend is smaller than zeta p this minimal backbend length is computed on a bethe lattice and shown to diverge as p pc 12 it is argued heuristically that on all lattices zeta p is proportional to the correlation length in the limit p to pc the chemical lengths of minimal backbend paths on the bethe lattice are calculated

percolation in cluster cluster aggregation processes numerical simulations of diffusion limited and reaction limited cluster cluster aggregation processes of identical particles are performed in a two dimensional box it is shown that for concentrations larger than a characteristic gel concentration the morphology of the resulting spanning cluster at the gel time t g exhibits a crossover length l c between percolation l l c and aggregation l c p the long range correlations are similar to that of percolation the vanishing links in the structure suggest that an homogeneous regime appears at small scales

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 discrete time inventory system with postponed demands in this article we consider a discrete time inventory model in which demands arrive according to a discrete markovian arrival process the inventory is replenished according to an s s policy and the lead time is assumed to follow a discrete phase type distribution the demands that occur during stock out periods either enter a pool which has a finite capacity n or leave the system with a predefined probability any demand that arrives when the pool is full and the inventory level is zero is assumed to be lost the demands in the pool are selected one by one if the on hand inventory level is above s 1 and the interval time between any two successive selections is assumed to have discrete phase type distribution the joint probability distribution of the number of customers in the pool and the inventory level is obtained in the steady state case the measures of system performance in the steady state are derived and the total expected cost rate is also calculated the results are illustrated numerically

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

computer simulation of a model for irreversible gelation a random mixture of bifunctional and tetrafunctional units is placed on a simple cubic lattice permanent bonds between these units are then formed by the random motion of active centres radicals resulting in a model for the gelation of polyacrylamide and similar processes the largest of the clusters formed by this kinetic percolation process is identified with the gel fraction the mean cluster size with the weight average degree of polymerisation the critical exponent gamma of the mean cluster size is roughly the same as for random percolation that is different from that of the classical theory of flory 1941 and stockmayer 1943 but the corresponding amplitude ratio that is the ratio of average molecular weight on both sides of the gel point differs strongly from its value in random percolation thus this kinetic gelation model seems to belong to a universality class of its own different from both that of random percolation and that of classical gelation theories

bridges in complex networks a bridge in a graph is an edge whose removal disconnects the graph and increases the number of connected components we calculate the fraction of bridges in a wide range of real world networks and their randomized counterparts we find that real networks typically have more bridges than their completely randomized counterparts but they have a fraction of bridges that is very similar to their degree preserving randomizations we define an edge centrality measure called bridgeness to quantify the importance of a bridge in damaging a network we find that certain real networks have a very large average and variance of bridgeness compared to their degree preserving randomizations and other real networks finally we offer an analytical framework to calculate the bridge fraction and the average and variance of bridgeness for uncorrelated random networks with arbitrary degree distributions

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

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

perimeter polynomials for bond percolation processes perimeter polynomials are given for the bond percolation problem on the following lattices the triangular up to d9 simple quadratic d13 honeycomb d17 face centred cubic d7 body centred cubic d8 simple cubic d9 and diamond d13 the total number of bond clusters grouped by size is given to two further orders in each case

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

the sol gel transition modelled by irreversible aggregation of clusters irreversible aggregation of clusters leads in a natural way to the formation of large ramified structures the authors investigate this growth process close to the sol gel transition where for the first time an infinite spanning cluster appears and find scaling behaviour the kinetics of the model leads to properties which are distinct from standard percolation two critical exponents are determined numerically the fractal dimension of the clusters is dg 1 75 or 0 07 in two dimensions

invading clusters in fractal media two dimensional invasion percolation simulations into regular fractals are performed to study the dependence of the invading cluster fractal dimension on the geometry of the medium the fractal dimension of the invading cluster di neither depends on the average coordination number of the network nor on the links of the backbone the authors find that di only depends on the fractal dimension of the backbone of the medium dbb and varies linearly with it

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

postbreakthrough behavior in flow through porous media we numerically simulate the traveling time of a tracer in convective flow between two points injection and extraction separated by a distance r in a model of porous media d 2 percolation we calculate and analyze the traveling time probability density function for two values of the fraction of connecting bonds p the homogeneous case p 1 and the inhomogeneous critical threshold case p p c we analyze both constant current and constant pressure conditions at p p c the homogeneous p 1 case serves as a comparison base for the more complicated p p c situation we find several regions in the probability density of the traveling times for the homogeneous case p 1 and also for the critical case p p c for both constant pressure and constant current conditions for constant pressure the first region i p corresponds to the short times before the flow breakthrough occurs when the probability distribution is strictly zero the second region ii p corresponds to numerous fast flow lines reaching the extraction point with the probability distribution reaching its maximum the third region iii p corresponds to intermediate times and is characterized by a power law decay the fourth region iv p corresponds to very long traveling times and is characterized by a different power law decaying tail the power law characterizing region iv p is related to the multifractal properties of flow in percolation and an expression for its dependence on the system size l is presented the constant current behavior is different from the constant pressure behavior and can be related analytically to the constant pressure case we present theoretical arguments for the values of the exponents characterizing each region and crossover times our results are summarized in two scaling assumptions for the traveling time probability density one for constant pressure and one for constant current we also present the production curve associated with the probability of traveling times which is of interest to oil recovery

potts model on finitely ramified fractals the author presents an exact solution for the ferromagnetic q state potts model on a family of finitely ramified fractal lattices with fractal dimension between 1 and 2 the author obtains the thermal exponent v for the whole fractal family the results stress the connection between geometrical parameters and critical behaviour opening up new possibilities for modelling physical systems showing fractal structures

scaling at the percolation threshold above six dimensions the fractal dimensionality of the infinite cluster at the percolation threshold for dimensionalities d 6 is shown to be d 4 rather than the naive finite size scaling prediction d d 2 similarly the conductivity of a sample of size l scales as l d rather than l 6 this anomalous behaviour is related to a dangerous irrelevant variable associated with the probability to have vertices of three bonds the crossover to the homogeneous behaviour occurs at length scales which are short compared with the correlation length the links and blobs picture is confirmed for d 6 and the size of the latter is estimated

extended eigenvalues for cesaro operators abstract a complex scalar is said to be an extended eigenvalue of a bounded linear operator t on a complex banach space if there is a non zero operator x such that t x x t such an operator x is called an extended eigenoperator of t corresponding to the extended eigenvalue the purpose of this paper is to give a description of the extended eigenvalues for the discrete cesaro operator c 0 the finite continuous cesaro operator c 1 and the infinite continuous cesaro operator c defined on the complex banach spaces l p l p 0 1 and l p 0 for 1 p by the expressions c 0 f n 1 n 1 k 0 n f k c 1 f x 1 x 0 x f t d t c f x 1 x 0 x f t d t it is shown that the set of extended eigenvalues for c 0 is the interval 1 for c 1 it is the interval 0 1 and for c it reduces to the singleton 1

invasion percolation a new form of percolation theory a new kind of percolation problem is described which differs from ordinary percolation theory in that it automatically finds the critical points of the system the model is motivated by the problem of one fluid displacing another from a porous medium under the action of capillary forces but in principle it may be applied to any kind of invasion process which proceeds along a path at least resistance the name invasion percolation is proposed for this new process similarities to and differences from ordinary percolation theory are discussed

a fluid eoq model of perishable items with intermittent high and low demand rates we consider a stochastic fluid eoq type model with demand rates operating in a two state random environment this environment alternates between exponentially distributed periods of high demand and generally distributed periods of low demand the inventory level starts at some level q and decreases linearly at rate h during the periods of high demand and at rate l h at periods of low demand refilling of the inventory level to level q is required when the first of two events takes place either the buffer level reaches zero or the buffer content becomes outdated if such an event occurs during a high demand period an order is instantaneously placed otherwise ordering is postponed until the beginning of the next high demand period we determine the steady state distribution of the inventory level as well as other quantities of interest such as the distribution of the time until a refill is required finally for a given cost revenue structure we determine the long run average profit and we co

convexity in hamilton jacobi theory i dynamics and duality value functions propagated from initial or terminal costs and constraints by way of a differential inclusion or more broadly through a lagrangian that may take on infty are studied in the case where convexity persists in the state argument such value functions themselves taking on infty are shown to satisfy a subgradient form of the hamilton jacobi equation which strongly supports properties of local lipschitz continuity semidifferentiability and clarke regularity an extended method of characteristics is developed which determines them from the hamiltonian dynamics underlying the given lagrangian close relations with a dual value function are revealed

compact operators without extended eigenvalues abstract a complex number is called an extended eigenvalue of a bounded linear operator t on a banach space b if there exists a non zero bounded linear operator x acting on b such that x t t x we show that there are compact quasinilpotent operators on a separable hilbert space for which the set of extended eigenvalues is the one point set 1

universality of cluster dynamics we have studied the kinetics of cluster formation for dynamical systems of dimensions up to n 8 interacting through elastic collisions or coalescence these systems could serve as possible models for gas kinetics polymerization and self assembly in the case of elastic collisions we found that the cluster size probability distribution undergoes a phase transition at a critical time which can be predicted from the average time between collisions this enables forecasting of rare events based on limited statistical sampling of the collision dynamics over short time windows the analysis was extended to l p normed spaces p 1 to allow for some amount of interpenetration or volume exclusion the results for the elastic collisions are consistent with previously published low dimensional results in that a power law is observed for the empirical cluster size distribution at the critical time we found that the same power law also exists for all dimensions n 2 8 two dimensional l p norms and even for coalescing collisions in two dimensions this broad universality in behavior may be indicative of a more fundamental process governing the growth of clusters

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

extended eigenvalues for bilateral weighted shifts abstract a complex scalar is said to be an extended eigenvalue for an operator a on a hilbert space h if there is a non zero operator x such that a x x a and in that case x is said to be an extended eigenoperator it is shown that if a bilateral weighted shift has a non unimodular extended eigenvalue then every extended eigenoperator for a is strictly lower triangular also it is shown that the set of the extended eigenvalues for an injective bilateral weighted shift is either c d or c 0 or d 0 or t and some examples are constructed in order to show that each of the four shapes does happen further it is shown that the set of the extended eigenvalues for an injective bilateral weighted shift with an even sequence of weights is either c 0 or t and that the set of the extended eigenvalues for an invertible bilateral weighted shift is t finally a factorization result is provided for the extended eigenoperators corresponding to a unimodular extended eigenvalue of an injective bilateral weighted shift

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

sensitivity analysis of optimal value functions of convex parametric programs with possibly empty solution sets let g y 1 y 2 be the optimal value of the abstract program which consists in minimizing a convex function f x to mathbb r cup infty over a feasible set of the form x in x a 1 x y 1 a 2 x leq k y 2 without assuming the existence of optimal solutions to this minimization problem we derive formulas for the subdifferential and the approximate subdifferential of g several applications are 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

subgradients of value functions in parametric dynamic programming we study in this paper the first order behavior of value functions in parametric dynamic programming with linear constraints and nonconvex cost functions by establishing an abstract result on the frechet subdifferential of value functions of parametric mathematical programming problems some new formulas on the frechet subdifferential of value functions in parametric dynamic programming are obtained

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

inhomogeneous percolation problems and incipient infinite clusters the authors consider inhomogeneous percolation models with density pc f x and examine the forms of f x which produce incipient structures taking f x approximately mod x mod lambda and assuming the existence of a correlation length exponent v for the homogeneous percolation model they prove that in d 2 the borderline value of lambda is lambda b 1 v if lambda 1 v then with probability one there is no infinite cluster while if lambda 2 the models studied suggest what sort of incipient objects should be examined in random surface models

topological and geometrical properties of random fractals the underlying structure of random fractals is described in terms of two exponents zeta and nu the exponent zeta is sensitive only to the topology or connectedness of the fractal whereas nu depends only on the geometry of the fractal the topological exponent arises from the scaling form for the distribution of paths nl on a finite fractal where a path is defined as the shortest walk from one site to another this path length distribution function is then used to derive expressions for the radius of gyration pair correlation function static structure factor and intensity of radiation scattered from percolation clusters at the gel point from these calculations the fractal dimension is shown to be zeta nu finally recent numerical results for percolation clusters percolation backbones and lattice animals are discussed

accessible external perimeters of percolation clusters the authors introduce a new definition of a cluster s accessible external perimeter which is different from the usual definition of the hull this external perimeter can be measured by probing adsorbent particles coming from the cluster s exterior the size of these particles determines which perimeter sites are accessible from the outside and hence one can define different external or accessible perimeters according to this size using a monte carlo simulation they measured the fractal dimensions of these accessible perimeters de for percolation clusters at the threshold on the square and the triangular lattices the fractal dimension of the accessible perimeter is found to depend on the size of the probing particles for several sizes which are larger than some lattice dependent threshold they find de approximately 4 3 which is significantly smaller than the hull dimension dh 7 4 found with smaller probing particles

sporadic and continuous clearing policies for a production inventory system under an m g demand process a production inventory system is filled continuously at rate 1 and satisfies demands of i i d random sizes that arrive at poisson times for this system we consider two clearing policies under sporadic review clearing takes place after a random time independent of the content process under continuous review the system is cleared as soon as the inventory level reaches some pre specified threshold we derive explicit results for the associated expected discounted cost functionals under both types of clearing

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

exact exponent relations for random resistor diode networks investigates the percolation properties of a random network of non directed bonds resistors and arbitrarily oriented directed bonds diodes for the square lattice there is a multicritical line which connects the isotropic percolation threshold with a network fully occupied by randomly oriented diodes random manhattan along this line symmetry and invariance properties are used to demonstrate that the correlation length exponents are constant furthermore a lattice independent relation is proved which shows that at the isotropic percolation threshold the correlation length diverges with the same exponent when the transition is approached by varying either the resistor concentration or the concentration of randomly oriented diodes

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

distributions and moments of structural properties for percolation clusters we study numerically and by scaling methods the distributions and moments of several structural properties of percolation clusters in two and three dimensions the clusters are generated at criticality and properties such as the distribution of the mass as a function of linear size or chemical distance are studied our results suggest that the hierarchy of moments can be represented by a single gap exponent using a scaling approach we obtain analytical forms for the different distribution functions which agree very well with the numerical data

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

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

optimality conditions under relaxed quasiconvexity assumptions using star and adjusted subdifferentials a set constrained optimization problem and a mathematical programming problem are considered we assume that the sublevel sets of the involving functions are convex only at the point under question and hence these functions are not assumed quasiconvex using the two star subdifferentials and the adjusted subdifferential we establish optimality conditions for usual minima and strict minima our results contain and improve some recent ones in the literature examples are provided to explain the advantages of each of our results

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

a connection between linear and nonlinear resistor networks the authors explore the connection between the higher moments of the current or voltage distribution in a random linear resistor network and the resistance of a nonlinear random resistor network they find that the two problems are very similar and that an infinite set of exponents are required to fully characterise each problem these exponent sets are shown to be identical on a particular hierarchical lattice a simple model which accurately describes the geometrical properties of the backbone of the infinite cluster at the percolation threshold and also the voltage distribution on this structure

structural crossover in dense irreversibly aggregating particulate systems cluster cluster aggregation has been simulated by off lattice monte carlo methods for diffusion limited dlca ballistic limited and reaction limited cluster aggregation classes we find that as the system evolves and becomes dense the largest cluster develops a hybrid structure with mass fractal dimension d f ensuremath simeq 2 6 over large length scales while at smaller length scales the early time dilute limit fractal structure is frozen in the largest cluster is thus an aggregate of smaller aggregates with a different fractal dimension and we call it a superaggregate the crossover length separating the two morphologies which we call the critical radius of gyration can be calculated based on a simple theory that assumes a monodisperse cluster size distribution this agrees well with simulation results for dlca however for other classes we find that the increasing polydispersity in cluster size pushes the simulated crossover length radius of gyration to values systematically larger than the predicted value

the tortuosity of occupied crossings of a box in critical percolation we consider the length of an occupied crossing of a box of size 0 n 0 3n d 1 in the short direction in standard bernoulli bond percolation on d at criticality let s n be the length of the shortest such crossing it is believed that s n 1 c in some sense for somec 0 here we show that if the correlation length p satisfies p c 1 n 1 logn 1 the assumption p c 3 p c p with 0 we also obtain a result about the length of optimal paths in first passage percolation

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

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

application of fractal concepts to polymer statistics and to anomalous transport in randomly porous media we consider the application of fractal concepts to polymer statistics and to anomalous transport in randomly porous media it is found that answers to interesting physics questions can be expressed in terms of several new fractal dimensions in addition to the fractal dimensiond f 1 d the fractal dimension of the backbone arises in connection with electric current flow 2 d red the fractal dimension of the singly connected bonds in the backbone arises in connection with its equivalence to the thermal scaling power 3 d e the fractal dimension of the of the elastic backbone 4 d u the fractal dimension of the unscreened perimeter arises in connection with the viscosity singularity at the gelation threshold 5 d min the fractal dimension of the minimum path or chemical distance between two sites arises in co nnection with the aharony stauffer conjecture 6 dw the fractal dimension of a random walk 7 d g the fractal dimension of growth sites that arise as a random walk creates a cluster relations among these fractal dimensions are discussed some of which can be proved and others of which are conjectures whose validity has been established only in certain limiting cases

scaling theory for the statistics of self avoiding walks on random lattices the authors study self avoiding walks saw on randomly diluted quenched lattices with direct configurational averaging over the moments of the saw distribution function a scaling function representation of rn the average end to end distance of n step walks is studied here both for saw on a the infinite percolation cluster and b any cluster they have shown that at the percolation threshold nu p nu p 1 beta p 2 nu p where beta p and nu p are the percolation order parameter and correlation length exponents respectively the authors also propose a scaling function representation for the total number of n step saw configuration gn approximately mu nngamma 1 for infinite cluster averaging which gives gamma p gamma d nu p nu for all cluster averaging gamma will remain unchanged

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