fractals in surface growth with power law noise the authors present a microscopic description of interface growth with power law noise distribution in the form p eta eta 1 approximately 1 mu which exhibits non universal roughening for the mu d 1 case in d 1 dimensions the existence of a fractal pattern in the bulk of the aggregate is explained leading trivially to the proof of the identity alpha z 2 for the roughening and the dynamical scaling exponents alpha and z respectively investigations on the distribution of step sizes of the discretized interface and the saturated growth speed further support the arguments

on the universality of crossing probabilities in two dimensional percolation six percolation models in two dimensions are studied percolation by sites and by bonds on square hexagonal and triangular lattices rectangles of widtha and heightb are superimposed on the lattices and four functions representing the probabilities of certain crossings from one interval to another on the sides are measured numerically as functions of the ratioa b in the limits set by the sample size and by the conventions and on the range of the ratioa b measured the four functions coincide for the six models we conclude that the values of the four functions can be used as coordinates of the renormalization group fixed point

stochastic renormalization group in percolation i fluctuations and crossover abstract a generalization of the renormalization group which describes order parameter fluctuations in finite systems is developed in the specific context of percolation this stochastic renormalization group srg expresses statistical self similarity through a non stationary branching process the srg provides a theoretical basis for analytical or numerical approximations both at and away from criticality whenever the correlation length is much larger than the lattice spacing regardless of the system size for example the srg predicts order parameter distributions and finite size scaling functions for the complete crossover between phases for percolation the simplest srg describes structural quantities conditional on spanning such as the total cluster mass or the minimum chemical distance between two boundaries in these cases the central limit theorem for independent random variables holds at the stable off critical fixed points while a fractal central limit theorem describing long range correlations holds at the unstable critical fixed point this first part of a series of articles explains these basic concepts and a general theory of crossover subsequent parts will focus on limit theorems and comparisons of small cell srg approximations with simulation results

homogenization and enhancement of the g equation in random environments we study the homogenization of a g equation which is advected by a divergence free stationary vector field in a general ergodic random environment we prove that the averaged equation is an anisotropic deterministic g equation and we give necessary and sufficient conditions in order to have enhancement since the problem is not assumed to be coercive it is not possible to have uniform bounds for the solutions in addition as we show the associated minimal first passage time function does not satisfy in general the uniform integrability condition which is necessary to apply the sub additive ergodic theorem we overcome these obstacles by i establishing a new reachability controllability estimate for the minimal function and ii constructing for each direction and almost surely a random sequence which has both a long time averaged limit due to the sub additive ergodic theorem and stays in the same sense asymptotically close to the minimal time

on the subdifferentials of quasiconvex and pseudoconvex functions and cyclic monotonicity abstract the notions of cyclic quasimonotonicity and cyclic pseudomonotonicity are introduced a classical result of convex analysis concerning the cyclic monotonicity of the fenchel moreau subdifferential of a convex function is extended to corresponding results for the clarke rockafellar subdifferential of quasiconvex and pseudoconvex functions the notion of proper quasimonotonicity is also introduced it is shown that this new notion retains the characteristic property of quasimonotonicity i e a lower semicontinuous function is quasiconvex if and only if its clarke rockafellar subdifferential is properly quasimonotone while it is also related to the kkm property of multivalued maps this makes it more useful in applications to variational inequalities

universality at the three dimensional percolation threshold the fraction of samples spanning a lattice at its percolation threshold is found via simulations of bond and site bond percolation to have a universal value of about 0 42 in three dimensions the ratio of shift of the finite to infinite size threshold to width of the threshold distribution is also compatible with universality lattices with up to 10013 sites and or very good statistics were needed to obtain clear results in these monte carlo tests preliminary studies with smaller lattices were inconclusive

test of universal finite size scaling in two dimensional site percolation traditional two scale factor universality concerning the number of clusters within a correlation volume near the percolation threshold is reconfirmed by a comparison of site percolation on square and triangular lattices we also test universality at the threshold e g of the ratio where is the correlation length and l the lattice size at the threshold the result is sensitive to the system s shape and aspect ratio boundary conditions the algorithm the detailed units measuring and l and possibly other factors as yet unexplored

amplitude ratios for critical systems in the c 2 universality class we study the finite size corrections of the critical dense polymer cdp and the dimer models on n rectangular lattice we find that the finite size corrections in the cdp and dimer models depend in a crucial way on the parity of n and a change of the parity of n is equivalent to the change of boundary conditions we present a set of universal amplitude ratios for amplitudes in finite size correction terms of critical systems in the universality class with central charge c 2 the results are in perfect agreement with a perturbated conformal field theory under the assumption that all analytical corrections coming from the operators which belongs to the tower of the identity our results inspire many interesting problems for further studies

lattice shapes and scaling functions for bond random percolation on honeycomb lattices we use a histogram monte carlo simulation method to calculate the scaling functions of the existence probability ep and the percolation probability p of the bond random percolation model on honeycomb lattices with aspect ratios of 0 5 and 2 0 we find that such different aspect ratios give quite different scaling functions near the critical region however they give a consistent critical point critical exponents and the thermodynamic order parameter from renormalization group calculations some interesting theoretical problems related to this work are discussed

the efficient determination of the percolation threshold by a frontier generating walk in a gradient the frontier in gradient percolation is generated directly by a type of self avoiding random walk the existence of the gradient permits one to generate an infinite walk on a computer of finite memory from this walk the percolation threshold pc for a two dimensional lattice can be determined with apparently maximum efficiency for a naive monte carlo calculation or n 12 for a square lattice the value pc 0 592745 or 0 000002 is found for a simulation of n 2 6 1011 total steps occupied and blocked perimeter sites the power of the method is verified on the kagome site percolation case

critical exponents of two dimensional potts and bond percolation models critical exponents of the two dimensional q state potts model are calculated by means of finite size scaling and transfer matrix techniques for continuous q results for the temperature exponent agree accurately with the conjecture of den nijs 1979 the magnetic exponent is found to behave in accordance with the conjecture of nienhuis et al 1980

boundary conditions and scaling functions of percolation models we use a histogram monte carlo simulation method to calculate the scaling functions of the existence probability ep and the percolation probability p of the site percolation model on square lattices with free and periodic boundary conditions we find that different boundary conditions give quite different scaling functions near the critical region however they give the consistent critical point critical exponents and the thermodynamic order parameter from renormalization group calculations similar results are found for other percolation models the implications of our calculated results for some theoretical problems of current interest are discussed

universality of critical existence probability for percolation on three dimensional lattices using a histogram monte carlo simulation method we calculate the existence probability for bond percolation on simple cubic sc and body centred cubic bcc lattices and site percolation on sc lattices with free boundary conditions the spanning rule considered by reynolds stanley and klein is used to define percolating clusters we find that for such systems has very good finite size scaling behaviour and the value of at the critical point is universal and is about

self similarity in the classification of finite size scaling functions for toroidal boundary conditions the conventional periodic boundary conditions in two dimensions are extended to general boundary conditions prescribed by primitive vector pairs that may not coincide with the coordinate axes this extension is shown to be unambiguously specified by the twisting scheme equivalent relations between different twist settings are constructed explicitly the classification of finite size scaling functions is discussed based on the equivalent relations a self similar pattern for distinct classes of finite size scaling functions is shown to appear on the plane that parametrizes the toroidal geometry

percolation fractals and the finite size scaling of existence probability abstract based on the connection between phase transitions of thermodynamic systems and percolation transitions we obtain the geometrical meaning of critical exponents using finite size scaling of the percolation probability p and the existence probability e p for percolating clusters and percolating subgraphs respectively where e p is the ratio of the total probability weight for percolating subgraphs to the total probability weight for all subgraphs we use a histogram monte carlo simulation method to calculate e p for a q state bond correlated percolation model qbcpm which is corresponding to the q state potts model on the square lattice with various linear dimensions we find that for a given q e p has very good scaling behavior but e p for different q has different scaling functions

scaling functions of the q state potts model on planar lattices abstract based on the subgraph expansion of the q state potts model qpm in an external field it has been shown that the qpm is corresponding to a q state bond correlated percolation model qbcpm the histogram monte carlo simulation method proposed by hu is used to calculate the existence probability e p g p q and the percolation probability p g p q of the qbcpm on the honeycomb the kagome and the plane triangular lattices with various linear dimensions from e p g p q and p g p q we obtain scaling functions of the qpm and qbcpm we find that as q or the coordination number of the lattices increases the widths of the scaling functions also increase the implication of this study is discussed

lower semicontinuous solutions of hamilton jacobi bellman equations the value function of mayer s problem arising in optimal control is investigated and lower semicontinuous solutions of the associated hamilton jacobi bellman equation are defined in three equivalent ways under quite weak assumptions about the control system the value function is the unique solution moreover it is stable with respect to perturbations of the control system and the cost it coincides with the viscosity solution whenever it is continuous

exact partition functions of the ising model on m n planar lattices with periodic aperiodic boundary conditions the grassmann path integral approach is used to calculate exact partition functions of the ising model on m n square sq plane triangular pt and honeycomb hc lattices with periodic periodic pp periodic antiperiodic pa antiperiodic periodic ap and antiperiodic antiperiodic aa boundary conditions the partition functions are used to calculate and plot the specific heat c kb as a function of temperature kbt j we find that for the n n sq lattice c kb for pa and ap boundary conditions are different from those for aa boundary conditions but for the n n pt and hc lattices c kb for ap pa and aa boundary conditions have the same values our exact partition functions might also be useful for understanding the effects of lattice structures and boundary conditions on critical finite size corrections of the ising model

a geometrical insight on pseudoconvexity and pseudomonotonicity generalised convexity is revisited from a geometric point of view a substitute to the subdifferential is proposed then generalised monotonicity is considered a representation of generalised monotone maps allows us to obtain a symmetry between maps and their inverses finally maximality of generalised monotone maps is analysed

the saga of a fish from a survival guide to closing lemmas abstract in the paper by d burago s ivanov and a novikov a survival guide for feeble fish it has been shown that a fish with limited velocity can reach any point in the possibly unbounded ocean provided that the fluid velocity field is incompressible bounded and has vanishing mean drift this result extends some known global controllability theorems though being substantially nonconstructive we give a fish a different recipe of how to survive in a turbulent ocean and show its relationship to structural stability of dynamical systems by providing a constructive way to change slightly the velocity field to produce conservative in the sense of not having wandering sets of positive measure dynamics in particular this leads to the extension of c pugh s closing lemma to incompressible vector fields over unbounded domains the results are based on an extension of the poincare recurrence theorem to some finite measures and on specially constructed newtonian potentials

universal scaling functions and quantities in percolation models abstract we briefly review recent work on universal finite size scaling functions ufssfs and quantities in percolation models the topics under discussion include a ufssfs for the existence probability also called crossing probability e p the percolation probability p and the probability w n of the appearance of n percolating clusters b universal slope for average number of percolating clusters c ufssfs for a q state bond correlated percolation model corresponding to the q state potts model we also briefly mention some very recent related developments and discuss implications of our results

minireview new results for old percolation abstract percolation theory deals with the formation of clusters in disordered media at the percolation threshold for the first time an infinite cluster appears however research of the last years showed that at this phase transition some old ideas were wrong there can also be two or three clusters spanning from top to bottom even in large two dimensional lattices and the probability of spanning is not given by simple real space renormalization ideas

on cardy s formula for the critical crossing probability in 2d percolation cardy s formula for the probability r of crossing a rectangular critical percolation system in the vertical direction with free boundaries on the two sides is written explicitly in terms of the aspect ratio r heightlwidth the first three terms are given by n 24 33i r 4 2 e r 3 e 7n 13 e 3zr 3 i which with n 1 nv i r m sufficient to determine x to high accuracy for all r