bifurcations of subharmonic solutions in reversible systems soit f r r n r r n une application c periodique cut on dit que l equation differentielle x f t x est reversible s il existe un operateur symetrique s sur r n tel que s 2 i de telles equations differentielles sont aussi appelees systemes avec la propriete e par rapport a s on etudie les bifurcations des solutions sous harmoniques de tels systemes reversibles

multiplicity of infinite clusters in percolation above six dimensions the author presents monte carlo simulations of site percolation near the percolation threshold in five to seven dimensions in contrast to lower dimensionalities more than one spanning cluster is found in a system of size ld for d 6 these results confirm coniglio s prediction 1985 that the breakdown of hyperscaling for d 6 is caused by a multiplicity of infinite clusters at the percolation threshold

the scaling limit of the incipient infinite cluster in high dimensional percolation ii integrated super brownian excursion for independent nearest neighbor bond percolation on zd with d 6 we prove that the incipient infinite cluster s two point function and three point function converge to those of integrated super brownian excursion ise in the scaling limit the proof is based on an extension of the new expansion for percolation derived in a previous paper and involves treating the magnetic field as a complex variable a special case of our result for the two point function implies that the probability that the cluster of the origin consists of n sites at the critical point is given by a multiple of n 3 2 plus an error term of order n 3 2 e with e 0 this is a strong version of the statement that the critical exponent is given by 2

coexistence of symmetric periodic points in the standard map the problem of the coexistence of symmetric periodic points of the standard map has been investigated first it is shown that there is another reversibility in addition to the one already known the dominant axis of this reversibility is the remaining coordinate axis using these reversibilities simple recursion formulas that determine the positions of symmetric periodic points have been derived second bifurcation structures of symmetric periodic points from the fixed point at the origin have been investigated it is shown that periodic points bifurcate along both the dominant axes of two reversibilities every time the rotation number of the fixed point passes through a rational number third the problem of the coexistence of symmetric periodic points has been numerically investigated with the aid of the recursion formulas a devil s staircase form of the evolution of the outermost kolmogorov arnold moser kam curve of the fixed point with the change of the perturbation parameter is observed this indicates that bifurcated periodic points do not disappear by inverse bifurcation as long as they are in the invariant region of the fixed point

error growth in the numerical integration of periodic orbits with application to hamiltonian and reversible systems we analyze in detail the growth with time of the coefficients of the asymptotic expansion of the error in the numerical integration with one step methods of periodic solutions of systems of ordinary differential equations variable stepsizes are allowed we successively consider general hamiltonian and reversible problems for hamiltonian and reversible systems and under fairly general hypotheses on the orbit being integrated numerical methods with relevant geometric properties symplecticness energy conservation reversibility are proved to have better error growth than general methods

the fractal dimension and other percolation exponents in four and five dimensions for percolation the fractal dimension df which is identical to the magnetic field scaling power yh has never been calculated for hypercubic lattices of dimension d 5 percolation is studied for systems of high dimensionality using the method of large cell monte carlo position space renormalisation group the estimate df yh 3 69 or 0 02 d 5 and 3 12 or 0 02 d 4 is obtained the thermal scaling power yt 1 nu where nu is the correlation length exponent is calculated yt 1 nu 0 51 or 0 05 d 5 and nu 0 64 or 0 02 d 4 the results are compared with the epsilon expansions of df and nu

periodic orbits in k symmetric dynamical systems abstract a map l is called k symmetric if itskth iterate lk possesses more symmetry than l for some value of k in k symmetric systems there exists a notion of k symmetric orbits this paper deels with k symmetric periodic orbits we derive a relation between orbits that are k symmetric with respect to reversing k symmetries and symmetric orbits of lk with this relation we set out an efficient method for finding systematically all periodic orbits that are k symmetric with respect to reversing k symmetries this k symmetric fixed set iteration fsi method generalizes a celebrated method due to devogelaere that applies to symmetric periodic orbits in reversible dynamical systems we use the fsi method to study k symmetric periodic orbits of a map of the planer2 possessing a crystallographic reversing k symmetry group the explicit findings illustrate a typically k symmetric phenomenon consisting of a nontrivial relation between the symmetry properties of periodic orbits and their periods

renormalisation in implicit complex maps abstract we study the dynamics of a family of implicit complex maps restricted to an invariant circle the renormalisation analysis for golden mean rotation number is given we find a one parameter family of universality classes and relate this to a line of fixed points of the square of the circle map renormalisation transformation these period 2 points are pairs of fractional linear maps

non algebraic integrability of the chew low reversible dynamical system of the cremona type and the relation with the 7th hilbert problem non resonant case abstract a smooth reversible dynamical system srds and a system of nonlinear functional equations defined by a certain rational quadratic cremona mapping and arising from the static model of the dispersion approach in the theory of strong interactions the chew low equations for p wave n scattering are considered this srds is splitted into 1 and 2 dimensional ones an explicit cremona transformation that completely determines the exact solution of the two dimensional system is found this solution depends on an odd function satisfying a nonlinear autonomous 3 point functional equation non algebraic integrability of srds under consideration is proved using the method of poincare normal forms and the siegel theorem on biholomorphic linearization of a mapping at a non resonant fixed point the proof is based on the classical feldman baker theorem on linear forms of logarithms of algebraic numbers which in turn relies upon solving the 7th hilbert problem by a i gel fond and t schneider and new powerful methods of a baker in the theory of transcendental numbers the general theorem following from the feldman baker theorem on applicability of the siegel theorem to the set of the eigenvalues c n of a mapping at a non resonant fixed point which belong to the algebraic number field a is formulated and proved the main results are presented in theorems 1 3 5 7 8 and remarks 3 7

stochastic approximations of constrained discounted markov decision processes abstract we consider a discrete time constrained markov decision process under the discounted cost optimality criterion the state and action spaces are assumed to be borel spaces while the cost and constraint functions might be unbounded we are interested in approximating numerically the optimal discounted constrained cost to this end we suppose that the transition kernel of the markov decision process is absolutely continuous with respect to some probability measure then by solving the linear programming formulation of a constrained control problem related to the empirical probability measure n of we obtain the corresponding approximation of the optimal constrained cost we derive a concentration inequality which gives bounds on the probability that the estimation error is larger than some given constant this bound is shown to decrease exponentially in n our theoretical results are illustrated with a numerical application based on a stochastic version of the beverton holt population model

critical percolation clusters in seven dimensions and on a complete graph we study critical bond percolation on a seven dimensional hypercubic lattice with periodic boundary conditions 7d and on the complete graph cg of finite volume number of vertices v we numerically confirm that for both cases the critical number density n s v of clusters of size s obeys a scaling form n s v s n over s v d f with identical volume fractal dimension d f 2 3 and exponent 1 1 d f 5 2 we then classify occupied bonds into bridge bonds which includes branch and junction bonds and nonbridge bonds a bridge bond is a branch bond if and only if its deletion produces at least one tree deleting branch bonds from percolation configurations produces leaf free configurations whereas deleting all bridge bonds leads to bridge free configurations composed of blobs it is shown that the fraction of nonbridge biconnected bonds vanishes n cg 0 for large cgs but converges to a finite value n 7d 0 0061931 7 for the 7d hypercube further we observe that while the bridge free dimension d bf 1 3 holds for both the cg and 7d cases the volume fractal dimensions of the leaf free clusters are different d lf 7d 0 669 9 2 3 and d lf cg 0 3337 17 1 3 on the cg and in 7d the whole leaf free and bridge free clusters all have the shortest path volume fractal dimension d min 1 3 characterizing their graph diameters we also study the behavior of the number and the size distribution of leaf free and bridge free clusters for the number of clusters we numerically find the number of leaf free and bridge free clusters on the cg scale as lnv while for 7d they scale as v for the size distribution we find the behavior on the cg is governed by a modified fisher exponent 1 while for leaf free clusters in 7d it is governed by fisher exponent 5 2 the size distribution of bridge free clusters in 7d displays two scaling behavior with exponents 4 and 1 the probability distribution p c 1 v dc 1 of the largest cluster of size c 1 for whole percolation configurations is observed to follow a single variable function p over x dx with x c 1 v d f for both cg and 7d up to a rescaling factor for the variable x the probability functions for cg and 7d collapse on top of each other within the entire range of x the analytical expressions in the x 0 and x limits are further confirmed our work demonstrates that the geometric structure of high dimensional percolation clusters cannot be fully accounted for by their complete graph counterparts

simplex algorithm for countable state discounted markov decision processes we consider discounted markov decision processes mdps with countably infinite state spaces finite action spaces and unbounded rewards typical examples of such mdps are inventory management and queueing control problems in which there is no specific limit on the size of inventory or queue existing solution methods obtain a sequence of policies that converges to optimality in value but may not improve monotonically ie a policy in the sequence may be worse than preceding policies our proposed approach considers countably infinite linear programming cilp formulations of the mdps a cilp is defined as a linear program lp with countably infinite numbers of variables and constraints under standard assumptions for analyzing mdps with countably infinite state spaces and unbounded rewards we extend the major theoretical extreme point and duality results to the resulting cilps under additional mild assumptions which are satisfied by several applications of interest we present a simplex type algor

from infinite to finite programs explicit error bounds with applications to approximate dynamic programming we consider linear programming lp problems in infinite dimensional spaces that are in general computationally intractable under suitable assumptions we develop an approximation bridge from the infinite dimensional lp to tractable finite convex programs in which the performance of the approximation is quantified explicitly to this end we adopt the recent developments in two areas of randomized optimization and first order methods leading to a priori as well as a posteriori performance guarantees we illustrate the generality and implications of our theoretical results in the special case of the long run average cost and discounted cost optimal control problems in the context of markov decision processes on borel spaces the applicability of the theoretical results is demonstrated through a fisheries management problem

time reversible and equivariant pitchfork bifurcation abstract in this note we describe a new phenomenon in steady state bifurcations of time reversible and equivariant vector fields when the eigenspace associated with a pair of imaginary eigenvalues that passes through zero onto the real line is separable from the nullspace associated with the fixed zeros of the spectrum we prove that a pitchfork bifurcation occurs

the bakerian lecture 1987 quantum chaology bounded or driven classical systems often exhibit chaos exponential instability that persists but their quantum counterparts do not nevertheless there are new regimes of quantum behaviour that emerge in the semiclassical limit and depend on whether the classical orbits are regular or chaotic and this motivates the following definition definition quantum chaology is the study of semiclassical but nonclassical behaviour characteristic of systems whose classical motion exhibits chaos this is illustrated by the statistics of energy levels on scales comparable with the mean level spacing of order h n for n freedoms these fall into universality classes for classically chaotic systems the statistics are those of random matrices real symmetric or complex hermitian depending on the presence or absence of time reversal symmetry for classically regular ones the statistics are poisson on larger scales of order h i e classically small but semiclassically large universality breaks down these phenomena are being explained by representing spectra in terms of classical closed orbits universal spectral behaviour has its origin in very long orbits non universal behaviour depends only on short ones

tupling in three dimensional reversible mappings feigenbaum s parameter scaling exponents s are calculated for l 3m l 5m l 7m l 9m sequences of periodic orbits in a class of three dimensional non measure preserving reversible mappings some preliminary results on orbit scaling exponents for l 3m sequences are also included scaling exponents found so far are the same as those found for two dimensional reversible mappings

an iterated implicit complex map abstract iterated complex maps defined by a root of a biquadratic equation g z z 0 are studied numerical studies suggest conjugacy functions with differentiable natural boundaries and maps having island chains occupying much of the riemann sphere an analogy with hamiltonian systems is discussed

exact results for an approximate renormalisation scheme and some predictions for the breakup of invariant ori abstract an approximate renormalisation scheme is derived for the breakup of invariant tori of arbitrary winding ratio in hamiltonian systems of one and a half degrees of freedom similar to that of escande and doveil it is a free semi group with two generators this scheme is solved exactly for its orbits stable manifolds unstable manifolds and critical set various results are found including a cantor set of universal fractal diagrams the robustness of noble tori and a scaling law for areas near critical circles

finite size effect in seven dimensional site percolation on a hypercubic lattice with l7 sites and l up to 21 the site percolation threshold for the spanning of a cluster from top to bottom is found to approach its asymptotic value with a 1 l law in contrast to the ising curie temperature in five dimensions

trace maps as 3d reversible dynamical systems with an invariant one link between the theory of quasicrystals and the theory of nonlinear dynamics is provided by the study of so called trace maps a subclass of them are mappings on a one parameter family of 2d surfaces that foliate 3 and also 3 they are derived from transfer matrix approaches to properties of 1d quasicrystals in this article we consider various dynamical properties of trace maps we first discuss the fibonacci trace map and give new results concerning boundedness of orbits on certain subfamilies of its invariant 2d surfaces we highlight a particular surface where the motion is integrable and semiconjugate to an anosov system i e the mapping acts as a pseudo anosov map we identify properties of symmetry and reversibility time reversal symmetry in the fibonacci trace map dynamics and discuss the consequences for the structure of periodic orbits we show that a conservative period boubling sequence can be identified when moving through the one parameter family of 2d surfaces by using generator trace maps in terms of which all trace maps obtained from invertible two letter substitution rules can be expressed we show that many features of the fibonacci trace map hold in general the role of the fricke character hat i x y z x 2 y 2 z 2 2xyz 1 its symmetry group and reversibility for the nielsen trace maps are described algebraically finally we outline possible higher dimensional generalizations

reversible systems with first integrals abstract a reversible system of ordinary differential equations ode possessing some first integrals is considered a simple description of the first integral space is given it is shown that under the auxiliary assumption that the integrals are symmetric i e they do not change sign under the action of the reversing involution the presence of the integrals allows one to bring the problem of dynamical stability for a symmetric equilibrium to the similar problem but for the reduced system the reduced system is merely the restriction of the initial system to the intersection of the surfaces of level of the integrals passing through the equilibrium

on the asymptotic optimality of finite approximations to markov decision processes with borel spaces calculating optimal policies is known to be computationally difficult for markov decision processes mdps with borel state and action spaces this paper studies finite state approximations of discrete time markov decision processes with borel state and action spaces for both discounted and average costs criteria the stationary policies thus obtained are shown to approximate the optimal stationary policy with arbitrary precision under quite general conditions for discounted cost and more restrictive conditions for average cost for compact state mdps we obtain explicit rate of convergence bounds quantifying how the approximation improves as the size of the approximating finite state space increases using information theoretic arguments the order optimality of the obtained convergence rates is established for a large class of problems we also show that as a pre processing step the action space can also be finitely approximated with sufficiently large number points thereby well known algorithms su

existence and stability of n pulses on optical fibers with phase sensitive amplifiers abstract the propagation of pulses in optical communication systems in which attenuation is compensated by phase sensitive amplifiers is investigated a central issue is whether optical fibers are capable of carrying several pieces of information at the same time in this paper multiple pulses are shown to exist for a fourth order nonlinear diffusion model due to kutz and co workers 1994 moreover criteria are derived for determining which of these pulses are stable the pulses arise in a reversible orbit flip a homoclinic bifurcation investigated here for the first time numerical simulations are used to study multiple pulses far away from the actual bifurcation point they confirm that properties of the multiple pulses including their stability are surprisingly well predicted by the analysis carried out near the bifurcation

coexistence of periodic points in reversible dynamical systems on a surface coexistence of periodic points is investigated in area preserving reversible dynamical systems on a plane which are not necessarily close to an integrable one it is shown that on the symmetry axis a higher cycle implies a lower cycle and any cycle implies an infinite number of higher cycles the standard map is investigated as an example it is shown that k cycles for any k 2 densely fill the phase plane of the map as the parameter goes to infinity in addition the number of these cycles are estimated

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

diffusion on random clusters and the parasite problem in the parasite problem a particle ant diffuses randomly on a random percolation cluster in the limit of concentration to 0 lattice animal monte carlo simulations and scaling arguments show that for large animals the distance r travelled by this parasite increases as t 1 za with time t the authors find za approximately 3 4 on the simple cubic lattice and za approximately 2 6 on the square lattice this anomalous diffusion is in rough agreement with a generalisation of a suggestion by alexander and orbach 1982 heuristic arguments in favour of this suggestion are given also they look at corrections to scaling for concentrations equal to the percolation threshold

the finite dimensional reversible kam theory abstract the main results of the theory of quasi periodic motions in finite dimensional reversible dynamical systems are surveyed lower dimensional invariant tori and excitation of elliptic normal modes are discussed in detail the case where the reversing involution is perturbed is considered several fundamental differences between the reversible and hamiltonian theories are listed

reversible long term integration with variable stepsizes the numerical integration of reversible dynamical systems is considered a backward analysis for variable stepsize one step methods is developed and it is shown that the numerical solution of a symmetric one step method implemented with a reversible stepsize strategy is formally equal to the exact solution of a perturbed differential equation which again is reversible this explains geometrical properties of the numerical flow such as the nearby preservation of invariants in a second part the efficiency of symmetric implicit runge kutta methods linear error growth when applied to integrable systems is compared with explicit nonsymmetric integrators quadratic error growth

existence of optimal policies for semi markov decision processes using duality for infinite linear programming semi markov decision processes on borel spaces with deterministic kernels have many practical applications particularly in inventory theory most of the results from general semi markov decision processes do not carry over to a deterministic kernel since such a kernel does not provide smoothness we develop infinite dimensional linear programming theory for a general stochastic semi markov decision process we give conditions general enough to allow deterministic kernels for solvability and strong duality of the resulting linear programs by using the developed linear programming theory we give conditions for the existence of a stationary deterministic policy for deterministic kernels which is optimal among all possible policies

generating functions for dynamical systems with symmetries integrals and differential invariants abstract we give a survey and some new examples of generating functions for systems with symplectic structure systems with a first integral systems that preserve volume and systems with symmetries and or time reversing symmetries both odes and maps are treated and we discuss how generating functions may be used in the structure preserving numerical integration of odes with the above properties

percolation in high dimensions is not understood abstract the number of spanning clusters in four to nine dimensions does not fully follow the expected size dependence for random percolation

invariant circles for the piecewise linear standard map we investigate invariant circles for a one parameter family of piecewise linear twist homeomorphisms of the annulus we show that invariant circles of all types and rotation numbers occur and we classify them into families we compute parameter ranges in which there are no invariant circles

droplet theory in low dimensions ising systems in zero field the authors develop a theory of the universal configurational physics underlying critical point phenomena in the ising universality class the theory is formally justifiable in d 1 epsilon dimensions and may be regarded as the natural continuation of the kink based theory of one dimension which it incorporates as a limiting case in d 1 epsilon the configurational building block is the droplet the typical droplet is not spherical and the many droplet assembly is not dilute the implied problems are handled with renormalisation group methods it is found that droplet shape fluctuation effects control the correlation length exponent nu while the nesting of droplets within droplets control the order parameter exponent beta the exponents nu and beta thus effectively define respectively the fractal dimensions of the droplet surface and the droplet volume the theory is used to determine and illuminate the critical behaviour of further quantities including the free energy the susceptibility the droplet number distribution and the distribution of the intra droplet order

the incipient infinite cluster for high dimensional unoriented percolation we consider bond percolation on mathbb z d at the critical occupation density p c for d 6 in two different models the first is the nearest neighbor model in dimension d 6 the second model is a spread out model having long range parameterized by l in dimension d 6 in the spread out case we show that the cluster of the origin conditioned to contain the site x weakly converges to an infinite cluster as x when d 6 and l is sufficiently large we also give a general criterion for this convergence to hold which is satisfied in the case d 6 in the nearest neighbor model by work of hara 12 we further give a second construction by taking p 6 the probability for invasion percolation to reach a site x is asymptotic to c x d 4 as x

period doubling in two parameter families abstract we study two aspects of the cross over from dissipative to area preserving behaviour for period doubling cascades first we consider the smoothness of the limiting bifurcation curve at the crossover point we make a conjecture and give supporting numerical evidence second we study the spectrum of the linearised doubling operator acting on arbitrary mappings and show that it can be expressed in terms of the spectrum of its restriction to area preserving mappings

passive scalars three dimensional volume preserving maps and chaos the dynamics of a medium sized particle passive scalar suspended in a general time periodic incompressible fluid flow can be described by three dimensional volume preserving maps in this paper these maps are studied in limiting cases in which some of the variables change very little in each iteration and others change quite a lot the former are called slow variables or actions the latter fast variables or angles the maps are classified by their number of actions for maps with only one action we find strong evidence for the existence of invariant surfaces that survive the nonlinear perturbation in a kam like way on the other hand for the two action case the motion is confined to invariant lines that break for arbitrary small size of the nonlinearity instead we find that adiabatic invariant surfaces emerge and typically intersect the resonance sheet of the fast motion at these intersections surfaces are locally broken and transitions from one to another can occur we call this process which is analogous to arnold diffusion singularity induced diffusion it is characteristic of two action maps in one action maps this diffusion is blocked by kam like surfaces

dual based methods for solving infinite horizon nonstationary deterministic dynamic programs we develop novel dual ascent and primal dual methods to solve infinite horizon nonstationary deterministic dynamic programs these methods are finitely implementable and converge in value to optimality moreover the dual ascent method produces a sequence of improving dual solutions that pointwise converge to an optimal dual solution while the primal dual algorithm provides a sequence of primal basic feasible solutions with value error bounds from optimality that converge to zero our dual based methods work on a more general class of infinite network flow problems that include the shortest path formulation of dynamic programs as a special case to our knowledge these are the first dual based methods proposed in the literature to solve infinite horizon nonstationary deterministic dynamic programs

poisson structure of the reversible 1 1 resonance abstract vector fields that are reversible but non hamiltonian are considered in the neighborhood of a 1 1 resonance it is found that the normalized vector field has a hidden poisson structure on the orbit space that is reminiscent of the equations governing the rigid body in mechanics the theory is used to give a geometric characterization of the phase space in the neighborhood of a reversible 1 1 resonance certain similarities and also essential differences with the hamiltonian 1 1 resonance are found

a new optimality criterion for nonhomogeneous markov decision processes we propose a new definition of optimality for nonhomogeneous markov decision processes called periodic forecast horizon pfh optimality using measures of discounting and ergodicity we establish conditions under which pfh optimal strategies exist and pfh optimality implies the more conventional notions of optimality and average optimality finally we use this definition of optimality as the basis for a direct development of forecast horizon results for discounted and undiscounted nonhomogeneous markov decision processes

time reversibility and particle sedimentation this paper studies an ordinary differential equation ode model called the stokeslet model and describes sedimentation of small clusters of particles in a highly viscous fluid this model has a trivial solution in which the n particles arrange themselves at the vertices of a regular n sided polygon when n 3 hocking j fluid mech 20 1964 pp 129 139 and caflisch et al phys fluids 31 1988 pp 3175 3179 prove the existence of periodic motion in the frame moving with the center of gravity in the cluster in which the particles form an isosceles triangle the study of periodic and quasiperiodic solutions of the stokeslet model is continued with emphasis on the spatial and time reversal symmetry of the model time reversibility is due to infinite viscosity and spatial dihedral symmetry is due to the assumption of identical particles and the symmetry of the trivial solution for three particles the existence of a second family of periodic solutions and a family of quasiperiodic solu

recognition of symmetries in reversible maps abstract we deal with germs of diffeomorphisms that are reversible under an involution we establish that this condition implies that in general both the family of reversing symmetries and the group of symmetries are not finite in contrast with continuous time dynamics where typically there are finitely many reversing symmetries from this we obtain two chains of fixed points subspaces of involutory reversing symmetries that we use to obtain geometric information on the discrete dynamics generated by a given diffeomorphism the results are illustrated by the generic case in arbitrary dimension when the diffeomorphism is the composition of transversal linear involutions

conditions for the existence of planning horizons we consider the general class of discounted problems involving sequential decision making over an unbounded horizon under the assumptions of a finite set of policy alternatives at each decision time and costs that are eventually uniformly bounded by some exponential existence of an optimal infinite horizon strategy is established moreover it is shown that there is a finite horizon version whose first policy decision agrees with an infinite horizon optimal policy under the additional assumption that the infinite horizon optimal strategy is eventually cyclic the stronger result follows that the infinite horizon optimal solution is unique for almost all interest rates and that therefore a planning horizon exists algorithmic implications are explored for a restricted subclass of these problems

mating quadratic maps with the modular group in 1994 s bullett and c penrose introduced the one complex parameter family of 2 2 holomorphic correspondences mathcal f a begin aligned left frac aw 1 w 1 right 2 left frac aw 1 w 1 right left frac az 1 z 1 right left frac az 1 z 1 right 2 3 end aligned and proved that for every value of a in 4 7 subset mathbb r the correspondence mathcal f a is a mating between a quadratic polynomial q c z z 2 c c in mathbb r and the modular group vargamma psl 2 mathbb z they conjectured that this is the case for every member of the family mathcal f a which has a in the connectedness locus we show here that matings between the modular group and rational maps in the parabolic quadratic family per 1 1 provide a better model we prove that every member of the family mathcal f a which has a in the connectedness locus is such a mating

an infinite dimensional linear programming algorithm for deterministic semi markov decision processes on borel spaces we devise an algorithm for solving the infinite dimensional linear programs that arise from general deterministic semi markov decision processes on borel spaces the algorithm constructs a sequence of approximate primal dual solutions that converge to an optimal one the innovative idea is to approximate the dual solution with continuous piecewise linear ridge functions that naturally represent functions defined on a high dimensional domain as linear combinations of functions defined on only a single dimension this approximation gives rise to a primal dual pair of semi infinite programs for which we show strong duality in addition we prove various properties of the underlying ridge functions

circumventing the slater conundrum in countably infinite linear programs abstract duality results on countably infinite linear programs are scarce subspaces that admit an interior point which is a sufficient condition for a zero duality gap yield a dual where the constraints cannot be expressed using the ordinary transpose of the primal constraint matrix subspaces that permit a dual with this transpose do not admit an interior point this difficulty has stumped researchers for a few decades it has recently been called the slater conundrum we find a way around this hurdle we propose a pair of primal dual spaces with three properties the series in the primal and dual objective functions converge the series defined by the rows and columns of the primal constraint matrix converge and the order of sums in a particular iterated series of a double sequence defined by the primal constraint matrix can be interchanged so that the dual is defined by the ordinary transpose weak duality and complementary slackness are then immediate instead of using interior point conditions to establish a zero duality gap we call upon the planning horizon method when the series in the primal and dual constraints are continuous we prove that strong duality holds if a sequence of optimal solutions to finite dimensional truncations of the primal and dual cilps has an accumulation point we show by counterexample that the requirement that such an accumulation point exist cannot be relaxed our results are illustrated using several examples and are applied to countable state markov decision processes and to a problem in robust optimization

a shadow simplex method for infinite linear programs we present a simplex type algorithm that is an algorithm that moves from one extreme point of the infinite dimensional feasible region to another not necessarily adjacent extreme point for solving a class of linear programs with countably infinite variables and constraints each iteration of this method can be implemented in finite time whereas the solution values converge to the optimal value as the number of iterations increases this simplex type algorithm moves to an adjacent extreme point and hence reduces to a true infinite dimensional simplex method for the important special cases of nonstationary infinite horizon deterministic and stochastic dynamic programs

euler poisson equations and existence of quasi periodic motions of heavy tops abstract we study the existence of quasi periodic motions of an asymmetric heavy top by using two different approaches the standard moser weinstein theorem and an extended liapounov center theorem we take advantage of the multiple reversibility of the euler poisson equations and are able to show that the extended liapounov center theorem yields more normal modes than does the moser weinstein theorem

a linear programming approach to nonstationary infinite horizon markov decision processes nonstationary infinite horizon markov decision processes mdps generalize the most well studied class of sequential decision models in operations research namely that of stationary mdps by relaxing the restrictive assumption that problem data do not change over time linear programming lp has been very successful in obtaining structural insights and devising solution methods for stationary mdps however an lp approach for nonstationary mdps is currently missing this is because the lp formulation of a nonstationary infinite horizon mdp includes countably infinite variables and constraints and research on such infinite dimensional lps has traditionally faced several hurdles for instance duality results may not hold an extreme point may not be a basic feasible solution and in the context of a simplex algorithm a pivot operation may require infinite data and computations and a sequence of improving extreme points need not converge in value to optimal in this paper we tackle these challenges a

on the numerical integration of ordinary differential equations by symmetric composition methods differential equations of the form dot x x a b are considered where the vector fields a and b can be integrated exactly enabling numerical integration of x by composition of the flows of a and b various symmetric compositions are investigated for order complexity and reversibility free lie algebra theory gives simple formulae for the number of determining equations for a method to have a particular order a new more accurate way of applying the methods thus obtained to compositions of an arbitrary first order integrator is described and tested the determining equations are explored and new methods up to 100 times more accurate at constant work than those previously known are given

period doubling bifurcations and universality in conservative systems we have studied numerically a sequence of period doubling bifurcations of periodic orbits of henon s conservative two dimensional mapping in agreement with other recent studies we also find evidence that such sequences possess universality properties similar to the ones observed for dissipative systems the corresponding universal constants however have significantly different values in each case we suggest that a possible explanation for this discrepancy may be the fact that conservative systems owing to their measure preserving property do not become asymptotically one dimensional as dissipative systems do

a tie breaking rule for discrete infinite horizon optimization we study discrete infinite horizon optimization problems without the common assumption of a unique optimum a method based on solution set convergence is employed for finding optimal initial decisions by solving finite horizon problems this method is applicable to general discrete decision models that satisfy a weak reachability condition the algorithm together with a stopping rule is applied to production planning and capacity expansion and computational results are reported

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

finite linear programming approximations of constrained discounted markov decision processes we consider a markov decision process mdp with constraints under the total expected discounted cost optimality criterion we are interested in proposing approximation methods of the optimal value of this constrained mdp to this end starting from the linear programming lp formulation of the constrained mdp on an infinite dimensional space of measures we propose a finite state approximation of this lp problem this is achieved by suitably approximating a probability measure underlying the random transitions of the dynamics of the system explicit convergence orders of the approximations of the optimal constrained cost are obtained by exploiting convexity properties of the class of relaxed controls we reduce the lp formulation of the constrained mdp to a finite dimensional static optimization problem that can be used to obtain explicit numerical approximations of the corresponding optimal constrained cost a numerical application illustrates our theoretical results

time reversal symmetry in dynamical systems a survey abstract in this paper we survey the topic of time reversal symmetry in dynamical systems we begin with a brief discussion of the position of time reversal symmetry in physics after defining time reversal symmetry as it applies to dynamical systems we then introduce a major theme of our survey namely the relation of time reversible dynamical sytems to equivariant and hamiltonian dynamical systems we follow with a survey of the state of the art on the theory of reversible dynamical systems including results on symmetric periodic orbits local bifurcation theory homoclinic orbits and renormalization and scaling some areas of physics and mathematics in which reversible dynamical systems arise are discussed in an appendix we provide an extensive bibliography on the topic of time reversal symmetry in dynamical systems

convergence of selections with applications in optimization we consider the problem of finding an easily implemented tie breaking rule for a convergent set valued algorithm i e a sequence of compact non empty subsets of a metric space converging in the hausdorff metric our tie breaking rule is determined by nearest point selections detined by uniqueness points in the space i e points having a unique best approximation in the limit set of the convergent algorithm convergence of the algorithm is shown to be equivalent to convergence of all such nearest point selections under reasonable additional hypotheses all points in the metric space have the uniqueness property consequently all points yield convergent nearest point selections i e tie breaking rules for a convergent algorithm we then show how to apply these results to approximate solutions for the following types of problems infinite systems of inequalities semi infinite mathematical programming non convex optimization and infinite horizon optimization

on universality for area preserving maps of the plane abstract we present detailed evidence that one parameter families of area preserving maps exhibit cascades of period doubling with universal geometric scaling in the parameter we relate this behaviour to a fixed point equation of the form 1 and det d 1 r 2 r 2 in particular we argue that the scaling transformation r 2 r 2 is conjugate to the transformation 0 x y x y with 2 and in fact 2 we present some numerical evidence that 8 721 1 4 018 1 16 36 where is the asymptotic ratio of the differences of the parameter values corresponding to the successive periods 2k described above

rolling horizon procedures in nonhomogeneous markov decision processes by far the most common planning procedure found in practice is to approximate the solution to an infinite horizon problem by a series of rolling finite horizon solutions although many empirical studies have been done this so called rolling horizon procedure has been the subject of few analytic studies we provide a cost error bound for a general rolling horizon algorithm when applied to infinite horizon nonhomogeneous markov decision processes both in the discounted and average cost cases we show that a doeblin coefficient of ergodicity acts much like a discount factor to reduce this error in particular we show that the error goes to zero for any fixed rolling horizon as this doeblin measure of control over the future decreases the theory is illustrated through an application to vehicle deployment

concepts of forecast and decision horizons applications to dynamic stochastic optimization problems in this paper we develop a rigorous new framework for the concepts of forecast and decision horizons these concepts are conditional in nature and in turn enable us to unify the existing concepts of strong and weak horizons moreover we are able to precisely state the questions of existence of forecast decision horizons finally we are able to further develop the theory of existence of forecast decision horizons in the discounted case and procedures to compute these horizons