finite time transport in aperiodic flows abstract we study the transport of particles in a general two dimensional incompressible flow in the presence of a transient eddy i e a bounded set of closed streamlines with a finite time of existence using quantities obtained from eulerian observations we provide explicit conditions for the existence of a hyperbolic structure in the flow which induces mixing between the eddy and its environment our results can be used directly to study finite time transport in numerically or experimentally generated vector fields with general time dependence

lagrangian coherent structures and mixing in two dimensional turbulence abstract we introduce a lagrangian definition for the boundaries of coherent structures in two dimensional turbulence the boundaries are defined as material lines that are linearly stable or unstable for longer times than any of their neighbors such material lines are responsible for stretching and folding in the mixing of passive tracers we derive an analytic criterion that can be used to extract coherent structures with high precision from numerical or experimental data sets the criterion provides a rigorous link between the lagrangian concept of hyperbolicity the okubo weiss criterion and vortex boundaries we apply the results to simulations of two dimensional barotropic turbulence

hyperbolic geometry of complex networks we develop a geometric framework to study the structure and function of complex networks we assume that hyperbolic geometry underlies these networks and we show that with this assumption heterogeneous degree distributions and strong clustering in complex networks emerge naturally as simple reflections of the negative curvature and metric property of the underlying hyperbolic geometry conversely we show that if a network has some metric structure and if the network degree distribution is heterogeneous then the network has an effective hyperbolic geometry underneath we then establish a mapping between our geometric framework and statistical mechanics of complex networks this mapping interprets edges in a network as noninteracting fermions whose energies are hyperbolic distances between nodes while the auxiliary fields coupled to edges are linear functions of these energies or distances the geometric network ensemble subsumes the standard configuration model and classical random graphs as two limiting cases with degenerate geometric structures finally we show that targeted transport processes without global topology knowledge made possible by our geometric framework are maximally efficient according to all efficiency measures in networks with strongest heterogeneity and clustering and that this efficiency is remarkably robust with respect to even catastrophic disturbances and damages to the network structure

finite size effects for percolation on apollonian networks we study the percolation problem on the apollonian network model the apollonian networks display many interesting properties commonly observed in real network systems such as small world behavior scale free distribution and a hierarchical structure by taking advantage of the deterministic hierarchical construction of these networks we use the real space renormalization group technique to write exact iterative equations that relate percolation network properties at different scales more precisely our results indicate that the percolation probability and average mass of the percolating cluster approach the thermodynamic limit logarithmically we suggest that such ultraslow convergence might be a property of hierarchical networks since real complex systems are certainly finite and very commonly hierarchical we believe that taking into account finite size effects in real network systems is of fundamental importance

convergence of threshold estimates for two dimensional percolation using a recently introduced algorithm for simulating percolation in microcanonical fixed occupancy samples we study the convergence with increasing system size of a number of estimates for the percolation threshold for an open system with a square boundary specifically for site percolation on a square lattice we show that the convergence of the average probability estimate is described by a nontrivial correction to scaling exponent as predicted previously and measure the value of this exponent to be 0 90 ifmmode pm else textpm fi 0 02 for the median and cell to cell estimates of the percolation threshold we verify that convergence does not depend on this exponent having instead a slightly faster convergence with a trivial analytic leading exponent

bounds of percolation thresholds in the enhanced binary tree it is argued that the lower critical percolation threshold of the enhanced binary tree is bounded as p c1 0 355 059 by studying its subgraphs while the upper threshold is bounded both from

optimal perturbation for enhanced chaotic transport the issue of determining the best perturbation which results in optimal chaotic flux across a separatrix is addressed using the melnikov function and lobe dynamics this theoretical analysis is motivated mainly through micro fluidic devices for which this problem has become important recently both two and three dimensional flows are analysed utilising a fourier transform representation the nature of the perturbation which maximises this flux for each frequency value is obtained the resulting optimally attainable flux is computed a concise bound on this flux is presented in terms of the supremum norm of the normal component of the perturbing velocity and the size of the heteroclinic manifold in this instance where the spatial part of the perturbation is permitted to be chosen based on the frequency it is shown that greater flux is achievable for smaller frequencies the theory is illustrated through two examples

erratum and addendum to a variational theory of hyperbolic lagrangian coherent structures physica d 240 2011 574 598 a b s t r a c t consider the dynamical system v x t x u r n t 1 where u is an open bounded subset of r n r is a bounded time interval and v u r n is a class c 3 vector field in its arguments let x t t0 x0 denote a trajectory of 1 passing the point x0 u at time t0 the flow map f t0 x0 takes an initial

ising models on hyperbolic graphs we consider ising models with ferromagnetic interactions and zero external magnetic field on the hyperbolic graph v f where v is the number of neighbors of each vertex and f is the number of sides of each face let tc be the critical temperature and t c sup t tc f 2 where f is the free boundary condition b c gibbs state is the plus b c gibbs state and is the minus b c gibbs state we prove that if the hyperbolic graph is self dual i e v f or if v is sufficiently large how large depends on f e g v 35 suffices for any f 3 and v 17 suffices for any f 17 then 0 t c tc in contrast with that t c tc for ising models on the hypercubic lattice zd with d 2 a result due to lebowitz 22 while whenever t t c f 2 the last result is an improvement in comparison with the analogous statement in refs 28 and 33 in which it was only proved that f 2 when t t c and it remains to show in both papers that f 2 whenever t t c therefore t c and tc divide 0 into three intervals 0 t c t c tc and tc in which but f 2 and f 2 and respectively

fast monte carlo algorithm for site or bond percolation we describe in detail an efficient algorithm for studying site or bond percolation on any lattice the algorithm can measure an observable quantity in a percolation system for all values of the site or bond occupation probability from zero to one in an amount of time that scales linearly with the size of the system we demonstrate our algorithm by using it to investigate a number of issues in percolation theory including the position of the percolation transition for site percolation on the square lattice the stretched exponential behavior of spanning probabilities away from the critical point and the size of the giant component for site percolation on random graphs

a unifying approximate dynamic programming model for the economic lot scheduling problem we formulate the well known economic lot scheduling problem elsp with sequence dependent setup times and costs as a semi markov decision process using an affine approximation of the bias function we obtain a semi infinite linear program determining a lower bound for the minimum average cost rate under a very mild condition we can reduce this problem to a relatively small convex quadratically constrained linear problem by exploiting the structure of the objective function and the state space this problem is equivalent to the lower bound problem derived by dobson dobson g 1992 the cyclic lot scheduling problem with sequence dependent setups oper res 40 736 749 and reduces to the well known lower bound problem introduced in bomberger bomberger ee 1966 a dynamic programming approach to a lot size scheduling problem management sci 12 778 784 for sequence dependent setups we thus provide a framework that unifies previous work and opens new paths for future research on tighter lower bounds

the average cost optimality equation for markov control processes on borel spaces abstract this paper deals with discrete time markov control processes with borel state space allowing unbounded costs and noncompact control sets for these models the existence of average optimal stationary policies has been recently established under very general assumptions using an optimality inequality here we give a condition which is a strengtened version of a variant of the vanishing discount factor approach for the optimality equation to hold

optimality inequalities for average cost markov decision processes and the stochastic cash balance problem for general state and action space markov decision processes we present sufficient conditions for the existence of solutions of the average cost optimality inequalities these conditions also imply the convergence of both the optimal discounted cost value function and policies to the corresponding objects for the average costs per unit time case inventory models are natural applications of our results we describe structural properties of average cost optimal policies for the cash balance problem an inventory control problem where the demand may be negative and the decision maker can produce or scrap inventory we also show the convergence of optimal thresholds in the finite horizon case to those under the expected discounted cost criterion and those under the expected discounted costs to those under the average costs per unit time criterion

average cost optimal policies for markov control processes with borel state space and unbounded costs abstract we show the existence of average cost optimal stationary policies for markov control processes with borel state space and unbounded costs per stage under a set of assumptions recently introduced by l i sennott 1989 for control processes with countable state space and finite control sets

a multiscale measure for mixing abstract we present a multiscale measure for mixing that is based on the concept of weak convergence and averages the mixedness of an advected scalar field at various scales this new measure referred to as the mix norm resolves the inability of the l 2 variance of the scalar density field to capture small scale variations when advected by chaotic maps or flows in addition the mix norm succeeds in capturing the efficiency of a mixing protocol in the context of a particular initial scalar field wherein lyapunov exponent based measures fail to do so we relate the mix norm to the classical ergodic theoretic notion of mixing and present its formulation in terms of the power spectrum of the scalar field we demonstrate the utility of the mix norm by showing how it measures the efficiency of mixing due to various discrete dynamical systems and to diffusion in particular we show that the mix norm can capture known exponential and algebraic mixing properties of certain maps we also analyze numerically the behaviour of scalar fields evolved by the standard map using the mix norm

ising model on a hyperbolic plane with a boundary a hyperbolic plane can be modeled by a structure called the enhanced binarytree we study the ferromagnetic ising model on top of the enhanced binarytree using the renormalization group analysis in

self avoiding walks and polygons on non euclidean lattices self avoiding walks and polygons on four non euclidean lattices are studied by the method of series expansions it appears that the polygons and walks have different critical points from each other and that both are in a different universality class to their cartesian lattice counterparts an exact solution is given for one lattice

absolute flux optimising curves of flows on a surface abstract given a flow on a surface we consider the problem of connecting two distinct trajectories by a curve of extremal absolute instantaneous flux we develop a complete classification of flux optimal curves accounting for the possibility of the flux having spatially and temporally varying weight this weight enables modelling the flux of non equilibrium distributions of tracer particles pollution concentrations or active scalar fields such as vorticity our results are applicable to all smooth autonomous flows area preserving or not

geometry of the ergodic quotient reveals coherent structures in flows abstract dynamical systems that exhibit diverse behaviors can rarely be completely understood using a single approach however by identifying coherent structures in their state spaces i e regions of uniform and simpler behavior we could hope to study each of the structures separately and then form the understanding of the system as a whole the method we present in this paper uses trajectory averages of scalar functions on the state space to a identify invariant sets in the state space and b to form coherent structures by aggregating invariant sets that are similar across multiple spatial scales first we construct the ergodic quotient the object obtained by mapping trajectories to the space of the trajectory averages of a function basis on the state space second we endow the ergodic quotient with a metric structure that successfully captures how similar the invariant sets are in the state space finally we parametrize the ergodic quotient using intrinsic diffusion modes on it by segmenting the ergodic quotient based on the diffusion modes we extract coherent features in the state space of the dynamical system the algorithm is validated by analyzing the arnold beltrami childress flow which was the test bed for alternative approaches the ulam s approximation of the transfer operator and the computation of lagrangian coherent structures furthermore we explain how the method extends the poincare map analysis for periodic flows as a demonstration we apply the method to a periodically driven three dimensional hill s vortex flow discovering unknown coherent structures in its state space finally we discuss differences between the ergodic quotient and alternatives propose a generalization to analysis of quasi periodic structures and lay out future research directions

average cost optimal stationary policies in infinite state markov decision processes with unbounded costs we deal with infinite state markov decision processes with unbounded costs three simple conditions based on the optimal discounted value function guarantee the existence of an expected average cost optimal stationary policy sufficient conditions are the existence of a distinguished state of smallest discounted value and a single stationary policy inducing an irreducible ergodic markov chain for which the average cost of a first passage from any state to the distinguished state is finite a result to verify this is also given two examples illustrate the ease of applying the criteria

zero sum stochastic games in borel spaces average payoff criteria this paper is concerned with two person zero sum dynamic stochastic games in borel spaces with possibly unbounded payoff function and several average or ergodic payoff criteria we give conditions under which the long run expected average payoff criterion the sample path average criterion the existence of solutions to the average payoff shapley equations and a certain martingale condition are all equivalent

robust markov control processes abstract we examine the average minimax control problem on a general state space with an unbounded cost function the controller is uncertain about his model in the sense that he deals with a range of alternative models as possibly true the aim of this paper is to establish the optimality inequality and to provide a minimax strategy for the controller a new feature in our approach is an application of a generalised tauberian relation that in turn allows us to impose relatively weak assumptions

upper transition point for percolation on the enhanced binary tree a sharpened lower bound hyperbolic structures are obtained by tiling a hyperbolic surface with negative gaussian curvature these structures generally exhibit two percolation transitions a system wide connection can be established at a certain occupation probability p p c1 and there emerges a unique giant cluster at p c2 g p c1 there have been debates about locating the upper transition point of a prototypical hyperbolic structure called the enhanced binary tree ebt which is constructed by adding loops to a binary tree this work presents its lower bound as p c2 ensuremath gtrsim 0 55 by using phenomenological renormalization group methods and discusses some solvable models related to the ebt

patchy percolation on a hierarchical network with small world bonds the bond percolation properties of the recently introduced hanoi networks are analyzed with the renormalization group unlike scale free networks these networks are meant to provide an analytically tractable interpolation between finite dimensional lattice based models and their mean field limits in percolation the hierarchical small world bonds in the hanoi networks impose order by uniting otherwise disconnected local clusters this patchy order results in merely a finite probability to obtain a spanning cluster for certain ranges of the bond probability unlike the usual 0 1 transition found on ordinary lattices the various networks studied here exhibit a range of phase behaviors depending on the prevalence of those long range bonds fixed points in general exhibit nonuniversal behavior

necessary and sufficient conditions for a bounded solution to the optimality equation in average reward markov decision chains abstract we consider average reward markov decision processes with discrete time parameter and denumerable state space we are concerned with the following problem find necessary and sufficient conditions so that for arbitrary bounded reward function the corresponding average reward optimality equation has a bounded solution this problem is solved for a class of systems including the case in which under the action of any stationary policy the state space is an irreducible positive recurrent class

crossing on hyperbolic lattices we divide the circular boundary of a hyperbolic lattice into four equal intervals and study the probability of a percolation crossing between an opposite pair as a function of the bond occupation probability p we consider the 7 3 heptagonal enhanced or extended binary tree ebt the ebt dual and the 5 5 pentagonal lattices we find that the crossing probability increases gradually from 0 to 1 as p increases from the lower p l to the upper p u critical values we find bounds and estimates for the values of p l and p u for these lattices and identify the self duality point p corresponding to where the crossing probability equals 1 2 comparison is made with recent numerical and theoretical results

scaling and universality in the spanning probability for percolation we discuss the spanning percolation probability function for three different spanning rules in general dimensions with both free and periodic boundary conditions our discussion relies on the renormalization group theory which indicates that apart from a few scale factors the scaling functions for the spanning probability are determined by the fixed point and therefore are universal for every system with the same dimensionality spanning rule aspect ratio and boundary conditions for square and rectangular systems we combine this theory with simple relations between the spanning rules and with duality arguments and find strong relations among different derivatives of the spanning function with respect to the scaling variables thus yielding several universal amplitude ratios and allowing a systematic study of the corrections to scaling both singular and analytic in the system size the theoretical predictions are numerically confirmed with excellent accuracy textcopyright 1996 the american physical society

majority vote model on hyperbolic lattices we study the critical properties of a nonequilibrium statistical model the majority vote model on heptagonal and dual heptagonal lattices such lattices have the special feature that they only ca

average cost markov decision processes with weakly continuous transition probabilities this paper presents sufficient conditions for the existence of stationary optimal policies for average cost markov decision processes with borel state and action sets and weakly continuous transition probabilities the one step cost functions may be unbounded and the action sets may be noncompact the main contributions of this paper are i general sufficient conditions for the existence of stationary discount optimal and average cost optimal policies and descriptions of properties of value functions and sets of optimal actions ii a sufficient condition for the average cost optimality of a stationary policy in the form of optimality inequalities and iii approximations of average cost optimal actions by discount optimal actions

reply to the comment on monte carlo simulation study of the two stage percolation transition in enhanced binary trees we discuss the nature of the two stage percolation transition on the enhanced binary tree in order to explain the disagreement in the estimation of the second transition probability between the one in our recent paper 2009 j phys a math theor 42 145001 and the other in the comment to it from baek et al 2009 j phys a math theor 42 478001 we point out some reasons that the finite size scaling analysis used by them is not proper for the enhanced tree due to its nonamenable nature which is verified by some numerical results

percolation on hyperbolic lattices the percolation transitions on hyperbolic lattices are investigated numerically using finite size scaling methods the existence of two distinct percolation thresholds is verified at the lower threshold an unbounded cluster appears and reaches from the middle to the boundary this transition is of the same type and has the same finite size scaling properties as the corresponding transition for the cayley tree at the upper threshold on the other hand a single unbounded cluster forms which overwhelms all the others and occupies a finite fraction of the volume as well as of the boundary connections the finite size scaling properties for this upper threshold are different from those of the cayley tree and two of the critical exponents are obtained the results suggest that the percolation transition for the hyperbolic lattices forms a universality class of its own

percolation in the hyperbolic plane the purpose of this paper is to study percolation in the hyperbolic plane and in transitive planar graphs that are quasi isometric to the hyperbolic plane

a counterexample on the optimality equation in markov decision chains with the average cost criterion abstract we consider markov decision processes with denumerable state space and finite control sets the performance index of a control policy is a long run expected average cost criterion and the cost function is bounded below for these models the existence of average optimal stationary policies was recently established in 11 under very general assumptions such a result was obtained via an optimality inequality here we use a simple example to prove that the conditions in 11 do not imply the existence of a solution to the average cost optimality equation

solutions of the average cost optimality equation for markov decision processes with weakly continuous kernel the fixed point approach revisited abstract this paper shows the existence of lower semicontinuous solutions of the average cost optimality equation for markov decision processes with borel spaces possible unbounded cost function and weakly continuous transition kernel this is done imposing a growth condition on the cost function a lyapunov stability condition on the transition kernel and a set of standard compactness continuity conditions the solution of the average cost optimality equation is obtained by means of the banach fixed point theorem

bounds of percolation thresholds on hyperbolic lattices we analytically study bond percolation on hyperbolic lattices obtained by tiling a hyperbolic plane with constant negative gaussian curvature the quantity of our main concern is p c2 the value of occupation probability where a unique unbounded cluster begins to emerge by applying the substitution method to known bounds of the order 5 pentagonal tiling we show that p c2 0 382508 for the order 5 square tiling p c2 0 472043 for its dual and p c2 0 275768 for the order 5 4 rhombille tiling

statistically optimal almost invariant sets abstract chaotic dynamical systems are often transitive although this transitivity is sometimes very weak it is of interest to divide the phase space into large regions between which there is relatively little communication of trajectories we present fast simple algorithms to find such divisions the present work builds on the results of froyland and dellnitz g froyland m dellnitz detecting and locating near optimal almost invariant sets and cycles siam j sci comput 24 6 2003 1839 1863 focussing on a statistical description of transitivity that takes into account the fact that trajectories tend to visit different regions of phase space with different frequencies the new work takes advantage of theoretical results from the theory of reversible markov chains a new adaptive algorithm is put forward to efficiently deal with situations where the boundaries of the weakly communicating regions are complicated this algorithm is illustrated with the standard map relevant convergence results are proven

a note on optimality conditions for continuous time markov decision processes with average cost criterion this note deals with continuous time markov decision processes with a denumerable state space and the average cost criterion the transition rates are allowed to be unbounded and the action set is a borel space we give a new set of conditions under which the existence of optimal stationary policies is ensured by using the optimality inequality our results are illustrated with a controlled queueing model moreover we use an example to show that our conditions do not imply the existence of a solution to the optimality equations in the previous literature

detecting and locating near optimal almost invariant sets and cycles the behaviors of trajectories of nonlinear dynamical systems are notoriously hard to characterize and predict rather than characterizing dynamical behavior at the level of trajectories we consider following the evolution of sets there are often collections of sets that behave in a very predictable way in spite of the fact that individual trajectories are entirely unpredictable such special collections of sets are invisible to studies of long trajectories we describe a global set oriented method to detect and locate these large dynamical structures our approach is a marriage of new ideas in modern dynamical systems theory and the novel application of graph dissection algorithms

definition and properties of lagrangian coherent structures from finite time lyapunov exponents in two dimensional aperiodic flows this paper develops the theory and computation of lagrangian coherent structures lcs which are defined as ridges of finite time lyapunov exponent ftle fields these ridges can be seen as finite time mixing templates such a framework is common in dynamical systems theory for autonomous and time periodic systems in which examples of lcs are stable and unstable manifolds of fixed points and periodic orbits the concepts defined in this paper remain applicable to flows with arbitrary time dependence and in particular to flows that are only defined computed or measured over a finite interval of time previous work has demonstrated the usefulness of ftle fields and the associated lcss for revealing the lagrangian behavior of systems with general time dependence however ridges of the ftle field need not be exactly advected with the flow the main result of this paper is an estimate for the flux across an lcs which shows that the flux is small and in most cases negligible for well defined lcss or those that rotate at a speed comparable to the local eulerian velocity field and are computed from ftle fields with a sufficiently long integration time under these hypotheses the structures represent nearly invariant manifolds even in systems with arbitrary time dependence the results are illustrated on three examples the first is a simplified dynamical model of a double gyre flow the second is surface current data collected by high frequency radar stations along the coast of florida and the third is unsteady separation over an airfoil in all cases the existence of lcss governs the transport and it is verified numerically that the flux of particles through these distinguished lines is indeed negligible

on the approximation of complicated dynamical behavior we present efficient techniques for the numerical approximation of complicated dynamical behavior in particular we develop numerical methods which allow us to approximate sinai ruelle bowen srb measures as well as almost cyclic behavior of a dynamical system the methods are based on an appropriate discretization of the frobenius perron operator and two essentially different mathematical concepts are used our idea is to combine classical convergence results for finite dimensional approximations of compact operators with results from ergodic theory concerning the approximation of srb measures by invariant measures of stochastically perturbed systems the efficiency of the methods is illustrated by several numerical examples

estimating long term behavior of flows without trajectory integration the infinitesimal generator approach the long term distributions of trajectories of a flow are described by invariant densities i e fixed points of an associated transfer operator in addition global slowly mixing structures such as almost invariant sets which partition phase space into regions that are almost dynamically disconnected can also be identified by certain eigenfunctions of this operator indeed these structures are often hard to obtain by brute force trajectory based analyses in a wide variety of applications transfer operators have proven to be very efficient tools for an analysis of the global behavior of a dynamical system the computationally most expensive step in the construction of an approximate transfer operator is the numerical integration of many short term trajectories in this paper we propose to directly work with the infinitesimal generator instead of the operator completely avoiding trajectory integration we propose two different discretization schemes a cell based discretization and a spectral coll

continuity of percolation probability on hyperbolic graphs let t be a ibrwarding tree of degree k where each vertex other than the origin has k children and one parent and the origin has k children but no parent k 2 define g to be the graph obtained by adding to t nearest neighbor bonds connecting the vertices which are in the same generation g is regarded as a discretization of the hyperbolic plane h in the same sense that z a is a discretization of r a independent percolation on g has been proved to have multiple phase transitions we prove that the percolation probability l p is continuous on 0 1 as a function ofp