the lusin area function and local admissible convergence of harmonic functions on homogeneous trees we prove admissible convergence to the boundary of functions that are harmonic on a subset of a homogeneous tree by means of a discrete green formula and an analogue of the lusin area function

spreading dynamics of forget remember mechanism we study extensively the forget remember mechanism frm for message spreading originally introduced in eur phys j b 62 247 2008 epjbfy1434 602810 1140 epjb e2008 00139 4 the freedom of specifying forget remember functions governing the frm can enrich the spreading dynamics to a very large extent the master equation is derived for describing the frm dynamics by applying the mean field techniques we have shown how the steady states can be reached under certain conditions which agrees well with the monte carlo simulations the distributions of forget and remember times can be explicitly given when the forget remember functions take linear or exponential forms which might shed some light on understanding the temporal nature of diseases like flu for time dependent frm there is an epidemic threshold related to the frm parameters we have proven that the mean field critical transmissibility for the sis model and the critical transmissibility for the sir model are the lower and the the upper bounds of the critical transmissibility for the frm model respectively

multiparameter maximal functions along dilation invariant hypersurfaces consider the hypersurface sn l ll x7 i in rn l the asso ciated maximal function operator is defined as the supremum of means taken over those parts of the surface lying above the rectangles 0 1 an analogous result is proved for a quadratic surface in r3

dependency of percolation critical exponents on the exponent of power law size distribution abstract the standard percolation theory uses objects of the same size moreover it has long been observed that the percolation properties of the systems with a finite distribution of sizes are controlled by an effective size and consequently the universality of the percolation theory is still valid in this study the effect of power law size distribution on the critical exponents of the percolation theory of the two dimensional models is investigated two different object shapes i e stick shaped and square are considered these two shapes are the representative of the fractures in fracture reservoirs and the sandbodies in clastic reservoirs the finite size scaling arguments are used for the connectivity to determine the dependency of the critical exponents on the power law exponent in particular the deviations of percolation exponents from their universal values as well as the connectivity behavior of such systems are investigated numerically as a result this extends the applicability of the conventional percolation approach to study the connectivity of systems with a very broad size distribution

on certain maximal functions and approach regions if f e l ir and u x y is the poisson integral of j a classical theorem of fatou 2 asserts that u has nontangential limits almost everywhere on r littlewood 5 and later zygmund 9 showed that this conclusion is false if nontangential approach is replaced by approach along translates of a curve which approaches the boundary tangentially in this paper we find a necessary and sufficient condition on an approach region for the associated maximal function to be appropriately bounded it then turns out that there are many approach regions which are not contained in any nontangential region but for which the conclusions of fatou s theorem remain true suppose q c ir has the property that whenever xi u e b and whenever x x y y it follows that x v e q then the maximal function associated to such a set is bounded if and only if the cross sectional measure at height y is bounded by a constant times y we also study regions where the cross sectional area is larger than y we introduce a modified maximal function which is again bounded this result leads to a simple proof of the boundedness of certain tangential maximal functions of poisson integrals of potentials studied in 161 the plan of this paper is as follows in section 1 we motivate our main ideas by discussing two particular examples the method of proof here is somewhat different from the proofs of the more general results in section 2 we obtain the necessary and sufficient conditions for fatou s theorem in section 3 we study more general approach regions and the appropriately modified maximal functions in section 4 we show how this maximal function controls the tangential boundary behavior of poisson integrals of potentials

percolation on correlated random networks we consider a class of random weighted networks obtained through a redefinition of patterns in an hopfield like model and by performing percolation processes we get information about topology and resilience properties of the networks themselves given the weighted nature of the graphs different kinds of bond percolation can be studied stochastic deleting links randomly and deterministic deleting links based on rank weights each mimicking a different physical process the evolution of the network is accordingly different as evidenced by the behavior of the largest component size and of the distribution of cluster sizes in particular we can derive that weak ties are crucial in order to maintain the graph connected and that when they are the most prone to failure the giant component typically shrinks without abruptly breaking apart these results have been recently evidenced in several kinds of social networks

critical exponents of the self avoiding walks on a family of finitely ramified fractals the authors have studied the self avoiding walks saw on a family of finitely ramified fractals the first member b 2 of the family is the two dimensional sierpinski gasket while the last member b infinity appears to be a wedge of the homogeneous triangular lattice by means of the exact renormalisation group transformations they have calculated the critical exponents alpha nu and gamma and the connectivity constant mu of saw on each member of a sequence 2 or b or 8 of the studied fractal family the obtained exact results are compared with the recent phenomenological proposals and with the results believed to be exact in the case of a homogeneous two dimensional lattice

on the critical behaviour of self avoiding walks for pt i see ibid vol 22 p 1839 1989 for the self avoiding walk problem the coefficients of the chain generating function and of the generating function for the sum of square end to end distances have been extended to 20 terms for the triangular lattice to 27 terms for the diamond lattice to 21 terms for the simple cubic lattice and to 16 terms for the bcc lattice precise estimates of the critical points are obtained and for the exponents we find that gamma 1 161 or 0 002 and nu 0 592 or 0 003 encompasses all the three dimensional lattice data

connective constant of saws on the sierpinski gasket family using a graph counting technique suitable for regular fractals an exact evaluation of the total number of embeddings of self avoiding walks on the generalized sierpinski gasket is obtained numerical estimates for the connective constants are quoted for the first time where b is the generation parameter of the gaskets it is shown that the number of distinct n step saws per site converges to the triangular lattice values when our analysis indicates that converges to the euclidean value in the same limit and an asymptotic expression is given

resilience to damage of graphs with degree correlations the existence or nonexistence of a percolation threshold on power law correlated graphs is a fundamental question for which a general criterion is lacking in this work we investigate the problems of site and bond percolation on graphs with degree correlations and their connection with spreading phenomena we obtain some general expressions that allow the computation of the transition thresholds or their bounds using these results we study the effects of assortative and disassortative correlations on the resilience to damage of networks

critical behavior of the susceptible infected recovered model on a square lattice by means of numerical simulations and epidemic analysis the transition point of the stochastic asynchronous susceptible infected recovered model on a square lattice is found to be c0 0 176 500 510 where c is the probability a chosen infected site spontaneously recovers rather than tries to infect one neighbor this point corresponds to an infection recovery rate of c 1 c0 c0 4 665 713 and a net transmissibility of 1 c0 1 3 c0 0 538 4102 which falls between the rigorous bounds of the site and bond thresholds the critical behavior of the model is consistent with the two dimensional percolation universality class but local growth probabilities differ from those of dynamic percolation cluster growth as is demonstrated explicitly

spreading of epidemics on scale free networks with nonlinear infectivity abstract in this paper we study the spreading of epidemics on scale free networks with infectivity which is nonlinear in the connectivity of nodes we will show that the nonlinear infectivity is more appropriate than constant or linear ones and give the epidemic threshold of the sis model on a scale free network with nonlinear infectivity in addition we compare the effects of nonlinear infectivity on the epidemic threshold with two other cases on infinite and finite scale free networks and find some new results such as with unit recovery rate and nonlinear irrational infectivity the epidemic threshold is always positive and the epidemic threshold can increase with network size on finite networks contrary to the findings in all previous work

geography in a scale free network model we offer an example of a network model with a power law degree distribution p k approximately k alpha for nodes but which nevertheless has a well defined geography and a nonzero threshold percolation probability for alpha 2 the range of real world contact networks this is different from p c 0 for alpha 3 results for the original well mixed scale free networks in our lattice based scale free network individuals link to nearby neighbors on a lattice even considerable additional small world links do not change our conclusion of nonzero thresholds when applied to disease propagation these results suggest that random immunization may be more successful in controlling human epidemics than previously suggested if there is geographical clustering

epidemics disorder and percolation abstract spatial models for spread of an epidemic may be mapped onto bond percolation we point out that with disorder in the strength of contacts between individuals patchiness in the spread of the epidemic is very likely and the criterion for epidemic outbreak depends strongly on the disorder because the critical region of the corresponding percolation model is broadened in some networks the percolation threshold is zero if another kind of disorder is present namely divergent fluctuations in the number of contacts we give an example a network with a well defined geography where this is not necessarily so and discuss whether real infection networks are likely to have this property

the susceptible infected recovered model on a euclidean network we consider a susceptible infected recovered sir epidemic model on a euclidean network in one dimension in which nodes at a distance l are connected with probability p l l in addition to nearest neighbors the topology of the network changes as is varied and its effect on the sir model is studied r t the recovered fraction of population up to time t and the total duration of the epidemic are calculated for different values of the infection probabilities q and a threshold behavior is observed for all up to 2 0 above the threshold value q qc the saturation value rsat attains a finite value both rsat and show scaling behavior in a finite system of size n and qc is constant for 0 qc the results are compared to static percolation phenomena and also to mean field results for finite systems discussions on the properties of the euclidean network are made in the light of the present results

on the boundary behaviour of generalized poisson integrals on symmetric spaces on a riemannian symmetric space x of the noncompact type we introduce a generalized poisson transformation from functions on the minimal boundary to functions on the maximal compactification whose restrictions to x are eigenfunctions of the invariant differential operators some continuityand fatou theorems are proved introduction let x g k be a riemannian symmetric space of the noncompact type by a theorem of kashiwara et al 5 every joint eigenfunction 4 of the invariant differential operators on x can be represented as a generalized poisson integral of a hyperfunction p on the minimal boundary k m and for generic eigenvalues p is the boundary value of 4 in a certain abstract sense when p is a continuous function p is the boundary value of 4 in a much more concrete way p is the limit along geodesics of the ratio 4 4x of 4 with the spherical function 4 a with the same eigenvalues as 4 this result holds under the condition on the eigenvalues that re x lies in the positive open weyl chamber and it is due to helgason 1 p 130 it was generalized to l functions qp on k m with pointwise almost everywhere admissible convergence by michelson 12 theorem 2 2 and to ll functions g on k m with pointwise almost everywhere restricted admissible convergence by sjogren 15 theorem 2 for 4 harmonic which is the same as x p these fatou theorems had been proved by helgason koranyi knapp williamson lindahl and stein 4 6 7 8 11 16 in this note we generalize these results of helgason michelson and sjogren to take into consideration the full boundary of x in the maximal satake furstenberg compactification x of x k m is only part of the boundary of x unless rank g k 1 it is the unique compact g orbit the other g orbits in the boundary can be identified with homogeneous spaces of the form g b where b is given by b m n k an for a parabolic subgroup p man of g for each such g orbit we define a generalized partial poisson transformation from functions on k m to sections of line bundles over g b we will prove that if qg is continuous the poisson received by the editors march 21 1984 and in revised form august 15 1984 1980 mathematics subject classification primary 43a85 secondary 53c35

clustering in complex networks ii percolation properties the percolation properties of clustered networks are analyzed in detail in the case of weak clustering we present an analytical approach that allows us to find the critical threshold and the size of the giant component numerical simulations confirm the accuracy of our results in more general terms we show that weak clustering hinders the onset of the giant component whereas strong clustering favors its appearance this is a direct consequence of the differences in the k core structure of the networks which are found to be totally different depending on the level of clustering an empirical analysis of a real social network confirms our predictions

admissible convergence of poisson integrals in symmetric spaces let f be an lp function p 1 on the distinguished boundary of a furstenberg satake compactification of a symmetric space we prove that the poisson integral pf of f converges admissibly at almost all components of each boundary of the compactification this was known previously only for large p in particular pf converges admissibly to f almost everywhere at the distinguished boundary a similar result is obtained for the normalized x poisson integral 9jf the method of proof uses maximal functions and it also gives a new proof of almost everywhere restricted convergence of pf and 9 xf for f e l

weighted inequalities and a e convergence for poisson integrals in light cones we show that the poisson maximal operator for the tube over the light cone p is bounded in the weighted space lp w if and only if the weight w x belongs to the muckenhoupt class ap we also characterize with a geometric condition related to the intrinsic geometry of the cone the weights v x for which p is bounded from lp v into lp u for some other weight u x 0 some applications to a e restricted convergence of poisson integrals are given