monte carlo study of the point for collapsing trees we investigate the collapse transition of lattice trees with nearest neighbor attraction in two and three dimensions two methods are used 1 a stochastic optimization process of the robbins monro type which is designed solely to locate the maximum value of the specific heat and 2 umbrella sampling which is designed to sample data over a wide temperature range as well as to combat the quasiergodicity of metropolis algorithms in the collapsed phase we find good evidence that the transition is second order with a divergent specific heat and that the divergence of the specific heat coincides with the metric collapse

density scaling a new monte carlo technique in statistical mechanics abstract we demonstrate the feasibility of using umbrella sampling to do monte carlo markov sampling runs each covering a substantial range of density density scaling monte carlo or dsmc one can obtain in this way not only the usual canonical averages but also the relative free energy as a function of density to test this it has been applied to systems for which there are some previous reliable results the hard sphere system and the restricted primitive model of 1 1 and 2 2 electrolytes the method proves to be startlingly powerful in that very extensive results can be obtained with very few dsmc runs an important further motivation is the prospect of using the technique to study phase transition regions

on the radius of convergence of interconnected analytic nonlinear input output systems a complete analysis is presented of the radii of convergence of the parallel product cascade and feedback interconnections of analytic nonlinear input output systems represented as fliess operators such operators are described by convergent functional series which are indexed by words over a noncommutative alphabet their generating series are therefore specified in terms of noncommutative formal power series given growth conditions for the coefficients of the generating series for the subsystems the radius of convergence of each interconnected system is computed assuming the subsystems are either all locally convergent or all globally convergent in the process of deriving the radius of convergence for the feedback connection it is shown definitively that local convergence is preserved under feedback this had been an open problem in the literature until recently

non gaussian limiting behavior of the percolation threshold in a large system we study short range percolation models in a finite box we define the percolation threshold as a random variable obtained from a stochastic procedure used in actual numerical calculations and study the asymptotic behavior of these random variables as the size of the box goes to infinity we formulate very general conditions under which in two dimensions rescaled threshold variables cannot converge to a gaussian and determine the asymptotic behavior of their second moments in terms of a widely used definition of correlation length we also prove that in all dimensions the finite volume percolation thresholds converge in probability to the percolation threshold of the infinite system the convergence result is obtained by estimating the rate of decay of the limiting distribution function s tail in terms of the correlation length exponent the proofs use exponential estimates of crossing probabilities substantial parts of the proofs apply in all dimensions

faa di bruno hopf algebra of the output feedback group for multivariable fliess operators abstract given two nonlinear input output systems written in terms of chen fliess functional expansions i e fliess operators it is known that the feedback interconnected system is always well defined and in the same class an explicit formula for the generating series of a single input single output closed loop system was provided by the first two authors in earlier work via hopf algebra methods this paper is a sequel it has four main innovations first the full multivariable extension of the theory is presented next a major simplification of the basic setup is introduced using a new type of grading that has recently appeared in the literature this grading also facilitates a fully recursive algorithm to compute the antipode of the hopf algebra of the output feedback group and thus the corresponding feedback product can be computed much more efficiently the final innovation is an improved convergence analysis of the antipode operation namely the radius of convergence of the antipode is computed

the scaling limit of self avoiding random walk in high dimensions the brydges spencer lace expansion is used to prove that the scaling limit of the finite dimensional distributions of self avoiding random walk in the d dimensional cubic lattice mathbb z d is gaussian if d is sufficiently large it is also shown that the critical exponent gamma for the number of self avoiding walks is equal to 1 if d is sufficiently large

virial coefficients for hard discs and hard spheres the sixth seventh and eighth virial coefficients of hard discs and hard spheres are evaluated numerically monte carlo integration i improve on the best previous estimates for the seventh virial coefficients and the integration of the eighth virial coefficient is new the best estimates for these coefficients for hard discs are b7 b6 0 114 86 7 and b8 b7 0 065 14 8 and for hard spheres b7 b6 0 01307 7 and b8 b7 0 00432 10 b is the second virial coefficient in each case pade approximations to the excess pressure and the excess free energy are computed from these results and compared to data otherwise obtained

on the critical percolation probabilities we prove that the critical probabilities of site percolation on the square lattice satisfy the relation p c p 1 furthermore we prove the continuity of the function percolation probability

correlation length and its critical exponent for percolation processes some critical exponent inequalities are given involving the correlation length of site percolation processes on d in particular it is shown thatv 2 d which implies that the critical exponentv cannot take its mean field value for the three dimensional percolation processes

a combinatorial hopf algebra for nonlinear output feedback control systems abstract in this work a combinatorial description is provided of a faa di bruno type hopf algebra which naturally appears in the context of fliess operators in nonlinear feedback control theory it is a connected graded commutative and non cocommutative hopf algebra defined on rooted circle trees a cancellation free forest formula for its antipode is given

technical communique extending the domain of definition of functional series for nonlinear systems functional series are the basis of much of existing nonlinear system theory the present note extends the well known local series beyond its radius of convergence to describe the entire system trajectory for as long as it exists first steps towards a generalization to systems defined on analytic manifolds are made

generating series for interconnected analytic nonlinear systems given two analytic nonlinear input output systems represented as fliess operators four system interconnections are considered in a unified setting the parallel connection product connection cascade connection and feedback connection in each case the corresponding generating series is produced and conditions for the convergence of the corresponding fliess operator are given in the process an existing notion of a composition product for formal power series has its set of known properties significantly expanded in addition the notion of a feedback product for formal power series is shown to be well defined in a broad context and its basic properties are characterized

a faa di bruno hopf algebra for a group of fliess operators with applications to feedback a faa di bruno type hopf algebra is developed for a group of integral operators known as fliess operators where operator composition is the group product such operators are normally written in terms of generating series over a noncommutative alphabet using a general series expansion for the antipode an explicit formula for the generating series of the compositional inverse operator is derived the result is applied to analytic nonlinear feedback systems to produce an explicit formula for the feedback product that is the generating series for the fliess operator representation of the closed loop system written in terms of the generating series of the fliess operator component systems this formula is employed to provide a proof that local convergence is preserved under feedback

percolation thresholds for discrete continuous models with nonuniform probabilities of bond formation this work has been partially supported by the iuventus plus programme founded by the polish ministry of science and higher education ip2014 024373 m l acknowledges spanish mineco project foqus fis2013 46768 erc adg osyris eu ip siqs eu strepequam and eu fetproact quic j w has been partially funded by nsf grant dms 131271

gap exponents for percolation processes with triangle condition a study is made of the gap exponents for percolation processes with the triangle condition in the subcritical region it is show that the gaps are given by t 2 fort 2 3 scaling theory predicts thatp p c 0 s p p c p ande p 1 c 0 c 0 s p p c p 3 wheres p is the typical cluster size it is found that p c p p p c 0 s p 1 p c p 1 2 and p c p 3 e p 1 c 0 c 0 s p 1 p c p 3 4

a unified approach to generating series for mixed cascades of analytic nonlinear input output systems the primary goal is to describe in a unified fashion the generating series for the cascade connection of any two analytic nonlinear input output systems in particular it will be shown that a single general definition of a composition product can be formulated in terms of formal power series to describe any possible cascade connection of analytic integral operators fliess operators and analytic functions provided that the composite system is well defined and resides in one of these two classes in each case the radius of convergence of the interconnection is computed

the distribution of percolating concentrations in finite systems abstract simulation of finite cubic lattices suggests that even in the limit of large systems the distribution of percolation thresholds deviates from a gaussian thus confirming ziff s two dimensional result

hyperscaling inequalities for percolation a set of critical exponent inequalities for independent percolation which saturate under the hyperscaling hypothesis is proved one of the consequences of the inequalities is the lower bounddc 6 for the upper critical dimension the proof is based on a rigorous version of the finite size scaling argument which extends easily to other systems such as ising ferromagnets

discrete time approximations of fliess operators a convenient way to represent a nonlinear input output system in control theory is via a chen fliess functional expansion or fliess operator the general goal of this paper is to describe how to approximate fliess operators with iterated sums and to provide accurate error estimates for two different scenarios one where the series coefficients are growing at a local convergence rate and the other where they are growing at a global convergence rate in each case it is shown that the error estimates are asymptotically achievable for certain worst case inputs the paper then focuses on the special case where the operators are rational i e they have rational generating series and thus are realizable in terms of bilinear ordinary differential state equations in particular it is shown that a discretization of the state equation via a kind of euler approximation coincides exactly with the discrete time fliess operator approximator of the continuous time rational operator

bilinear system interconnections and generating series of weighted petri nets this paper has three objectives the first objective is to provide a simpler proof concerning a sufficient condition due to ferfera under which bilinearity is preserved for cascade interconnections the next objective is to provide a specific counterexample to show that this condition does not apply to the feedback connection the final objective is to show that the well known correspondence between rational series and formal power series recognized by weighted finite state automata can be generalized to produce a correspondence between the generating series of cascaded and feedback connected bilinear systems and a class of weighted petri nets

the probability distribution of the percolation threshold in a large system we show that the distribution of the percolation threshold in a large finite system does not converge to a gaussian when the size of the system goes to infinity provided that the two widely accepted definitions of correlation length are equivalent the shape of the distribution is thus directly related to the presence or absence of logarithmic corrections in the power law for the correlation length the result is obtained by estimating the rate of decay of tail of the limiting distribution in terms of the correlation length exponent v all results are rigorously proven in the 2d case generalizations for three dimensions are also discussed

fliess operators on lp spaces convergence and continuity fliess operators as input output mappings are particularly useful in a number of fundamental problems concerning nonlinear realization theory in the classical analysis of these operators certain growth conditions on the coe3cients in their series representations insure uniform and absolute convergence provided every input is uniformly bounded by some 4xed upperbound in some emerging applications however it is more natural to consider other classes of inputs in this paper lp function spaces are considered in particular it is shown that the classic growth conditions also provide su3cient conditions for convergence and continuity when the admissible inputs are from a ball in lp t0 t 0 t where t is bounded and p 1 in addition stronger global growth conditions are given that apply even for the case where t is unbounded when the coe3cients of a fliess operator have a state space representation it is shown that the state space model will always locally realize the correspondinginput output map on lp t0 t 0 t for su3ciently small t 0 if certain well posedness conditions are satis4ed then the state space model will globally realized the input output mapping for unbounded t when the coe3cients satisfy the global growth condition c 2002 elsevier science b v all rights reserved

left inversion of analytic nonlinear siso systems via formal power series methods given a single input single output siso system f and a function y in the range of f the left inversion problem is to determine an input u such that y f u the goal of this paper is to provide an exact and explicit analytical solution to this problem in the case where f is an analytic mapping in the sense that it has a convergent chen fliess functional expansion and y is a real analytic function in particular it will be shown that given a certain condition on the generating series c of f a corresponding unique analytic u can always be determined via operations on formal power series the condition on c turns out to be equivalent to having a well defined relative degree when f has an input affine analytic state space realization with finite dimension but the method is applicable even when f does not have such a realization the technique is demonstrated on four examples including a continuous stirred chemical reactor

mean field critical behaviour for percolation in high dimensions the triangle condition for percolation states that sum limits x y tau 0 x tau 0 y cdot tau y 0 is finite at the critical point where x y is the probability that the sitesx andy are connected we use an expansion related to the lace expansion for a self avoiding walk to prove that the triangle condition is satisfied in two situations i for nearest neighbour independent bond percolation on thed dimensional hypercubic lattice ifd is sufficiently large and ii in more than six dimensions for a class of spread out models of independent bond percolation which are believed to be in the same universality class as the nearest neighbour model the class of models in ii includes the case where the bond occupation probability is constant for bonds of length less than some large number and is zero otherwise in the course of the proof an infrared bound is obtained the triangle condition is known to imply that various critical exponents take their mean field bethe lattice values gamma beta 1 delta delta t 2 t underset raise0 3em hbox smash scriptscriptstyle geqslant 2 and that the percolation density is continuous at the critical point we also prove thatv2 in i and ii wherev2 is the critical exponent for the correlation length

are the tails of percolation thresholds gaussians the probability distribution of percolation thresholds in finite lattices was first believed to follow a normal gaussian behaviour with increasing computer power and more efficient simulational techniques this belief turned to a stretched exponential behaviour instead here based on a further improvement of the monte carlo data we show evidence on square lattices that this question is numerically not yet answered at all