the dirichlet problem for nonlinear second order elliptic equations iii functions of the eigenvalues of the hessian on etudie le probleme de dirichlet dans un domaine borne de r n a frontiere lisse f d 2 u dans u sur

quantum monte carlo methods recent developments abstract recent developments of the quantum monte carlo methods are reviewed briefly with emphasis on basic ideas namely the transformation of a d dimensional quantum system to the corresponding d 1 dimensional classical system some recent applications to quantum spin systems are summarized in particular numerical evidence of the haldane gap is given the quantum transfer matrix method is also presented for it is also based on the st transformation the quantum monte carlo cam is explained briefly fractal path integrals are also presented namely a general theory of decomposition of exponential operators and symplectic integrators are explained the negative sign problem is also discussed

classification of phase transitions in reaction diffusion models equilibrium phase transitions are associated with rearrangements of minima of a lagrangian potential treatment of nonequilibrium systems requires doubling of degrees of freedom which may be often interpreted as a transition from the coordinate to the phase space representation as a result one has to deal with the hamiltonian formulation of the field theory instead of the lagrangian one we suggest a classification scheme of phase transitions in reaction diffusion models based on the topology of the phase portraits of corresponding hamiltonians in models with an absorbing state such a topology is fully determined by intersecting curves of zero energy we identify four families of topologically distinct classes of phase portraits stable upon renormalization group transformations

regularity of potential functions of the optimal transportation problem the potential function of the optimal transportation problem satisfies a partial differential equation of monge ampere type in this paper we introduce the notion of a generalized solution and prove the existence and uniqueness of generalized solutions of the problem we also prove the solution is smooth under certain structural conditions on the cost function

decomposition formulas of exponential operators and lie exponentials with some applications to quantum mechanics and statistical physics decomposition formulas of general exponential operators in a banach algebra and in a lie algebra are presented that yield a basis of monte carlo simulation of quantum systems they are applied to study the relaxation and fluctuation from the initially unstable point and to confirm algebraically the scaling theory of transient phenomena a global approximation method of transient phenomena is also formulated on the basis of decomposition formulas it is applied to the laser model as a simple example

numerical study of the langevin theory for fixed energy sandpiles the recently proposed langevin equation aimed to capture the relevant critical features of stochastic sandpiles and other self organizing systems is studied numerically the equation is similar to the reggeon field theory describing generic systems with absorbing states but it is coupled linearly to a second conserved and static nondiffusive field it has been claimed to represent a different universality class including different discrete models the manna as well as other sandpiles reaction diffusion systems etc in order to integrate the equation and surpass the difficulties associated with its singular noise we follow a numerical technique introduced by dickman our results coincide remarkably well with those of discrete models claimed to belong to this universality class in one two and three dimensions this provides a strong backing for the langevin theory of stochastic sandpiles and to the very existence of this meagerly understood universality class

learnability of quantified formulas we consider the following classes of quantified formulas fix a set of basic relations called a basis take conjunctions of these basic relations applied to variables and constants in arbitrary ways finally quantify existentially or universally some of the variables we introduce some conditions on the basis that guarantee efficient learnability furthermore we show that with certain restrictions on the basis the classification is complete we introduce as an intermediate tool a link between this class of quantified formulas and some well studied structures in universal algebra called clones more precisely we prove that the computational complexity of the learnability of these formulas is completely determined by a simple algebraic property of the basis of relations their clone of polymorphisms finally we use this technique to give a simpler proof of the already known dichotomy theorem over boolean domains

integration of langevin equations with multiplicative noise and the viability of field theories for absorbing phase transitions efficient and accurate integration of stochastic partial differential equations with multiplicative noise can be obtained through a split step scheme which separates the integration of the deterministic part from that of the stochastic part the latter being performed by sampling exactly the solution of the associated fokker planck equation we demonstrate the computational power of this method by applying it to the most absorbing phase transitions for which langevin equations have been proposed this provides precise estimates of the associated scaling exponents clarifying the classification of these nonequilibrium problems and confirms or refutes some existing theories

the crossover phenomena in surface growth models with height dependent noise abstract in this paper we consider several known growth processes with height dependent noise this type of noise is interesting from a theoretical standpoint for example it paves the way to the derivation of the exact height distribution of the kpz equation through the hopf cole transformation in addition it may have implications for experimental growth processes using numerical methods we observe that adding such a noise to different growth processes can change their universality class or ruin the scaling laws in the case of mullins herring equation a two fold cross over is observed

universality in stochastic exponential growth recent imaging data for single bacterial cells reveal that their mean sizes grow exponentially in time and that their size distributions collapse to a single curve when rescaled by their means an analogous result holds for the division time distributions a model is needed to delineate the minimal requirements for these scaling behaviors we formulate a microscopic theory of stochastic exponential growth as a master equation that accounts for these observations in contrast to existing quantitative models of stochastic exponential growth e g the black scholes equation or geometric brownian motion our model the stochastic hinshelwood cycle shc is an autocatalytic reaction cycle in which each molecular species catalyzes the production of the next by finding exact analytical solutions to the shc and the corresponding first passage time problem we uncover universal signatures of fluctuations in exponential growth and division the model makes minimal assumptions and we describe how more complex reaction networks can reduce to such a cycle we thus expect similar scalings to be discovered in stochastic processes resulting in exponential growth that appear in diverse contexts such as cosmology finance technology and population growth

the magnus expansion for classical hamiltonian systems lie operator techniques and the magnus expansion are developed in the framework of classical mechanics this leads to an exponential perturbation theory that preserves the canonical character at each order of approximation the treatment is kept as close as possible to the quantum mechanical case in order take full advantage of the properties of the expansion the explicit relationship with secular perturbation theory is established and a recursive procedure for obtaining higher order approximants is provided finally the formalism is applied to two problems of physical interest

the dirichlet problem for nonlinear second order elliptic equations i monge ampegre equation on considere le probleme de dirichlet pour des equations elliptiques non lineaires pour une fonction reelle u definie dans la fermeture d un domaine borne dans r n avec une frontiere c

regularity for an obstacle problem of hessian equations on riemannian manifolds abstract in this paper we study the regularity for solutions to an obstacle problem of hessian type fully nonlinear equations on riemannian manifolds as an application the existence of a c 1 1 solution is proved

study of correction terms for higher order decompositions of exponential operators many kinds of higher order decompositions are applied to an exactly solvable model and the trotter number dependence of the correction terms is evaluated systematically practically useful decompositions are explored

confirming and extending the hypothesis of universality in sandpiles stochastic sandpiles self organize to an absorbing state critical point with scaling behavior different from directed percolation dp and characterized by the presence of an additional conservation law this is usually called the c dp or manna universality class there remains however an exception to this universality principle a sandpile automaton introduced by maslov and zhang which was claimed to be in the dp class despite the existence of a conservation law we show by means of careful numerical simulations as well as by constructing and analyzing a field theory that contrarily to what was previously thought this sandpile is also in the c dp or manna class this confirms the hypothesis of universality for stochastic sandpiles and gives rise to a fully coherent picture of self organized criticality in systems with conservation in passing we obtain a number of results for the c dp class and introduce a strategy to easily discriminate between dp and c dp scaling

fractal path integrals with applications to quantum many body systems abstract a general formulation of the fractal decomposition of exponential operators is reviewed briefly and its mathematical structure is discussed together with some applications to quantum many body systems more explicitly the operator e x a b is decomposed into a product of the form ex a b f m x o x m 1 with f m x e t 1 xa e f 2 xb e f 3 xa e t 4 xb e t m xb the parameters t j show a fractal structure as shown in the text

free boundaries in optimal transport and monge ampere obstacle problems given compactly supported 0 f g 2 l 1 r n the problem of trans porting a fraction m min kfkl1 kgkl1 of the mass of f onto g as cheaply as possible is considered where cost per unit mass transported is given by a cost function c typically quadratic c x y x y 2 2 this question is shown to be equivalent to a double obstacle problem for the monge ampere equation for which sucient conditions are given to guarantee uniqueness of the solution such as f vanishing on sptg in the quadratic case the part of f to be transported increases monotonically with m and if sptf and sptg are separated by a hyperplane h then this part will separated from the balance of f by a semiconcave lipschitz graph over the hyperplane if f f and g g are bounded away from zero and infinity on separated strictly convex domains r n for the quadratic cost this graph is shown to be a c 1 loc hypersurface in whose normal coincides with the direction transported the optimal map between f and g is shown to be holder continuous up to this free bound ary and to those parts of the fixed boundary which map to locally convex parts of the path connected target region

numerical analysis of a langevin equation for systems with infinite absorbing states one dimensional systems with an infinite number of absorbing states exhibit a phase transition that is not fully understood yet their static critical exponents are universal and belong in the reggeon field theory or directed percolation universality class however exponents associated with the spreading of a localized seed appear to be nonuniversal depending on the nature of the initial condition we investigate this problem by integrating numerically a non markovian langevin equation proposed recently to describe such phase transitions we find that the static critical exponents are universal as expected on the other hand the langevin equation reproduces the nonuniversal behavior observed in microscopic models for exponents associated with the spreading of an initially localized seed and satisfies the generalized hyperscaling relation proposed for those systems

hyperscaling in the domany kinzel cellular automaton an apparent violation of hyperscaling at the endpoint of the critical line in the domany kinzel stochastic cellular automaton finds an elementary resolution upon noting that the order parameter is discontinuous at this point we derive a hyperscaling relation for such transitions and discuss applications to related examples

stability of the trotter suzuki decomposition the trotter suzuki decomposition tsd is an important tool for the simulation and control of physical systems we provide evidence for the stability of the tsd we model the error in the decomposition and determine sufficiency conditions that guarantee the stability of this decomposition under this model we relate these sufficiency conditions to precision limitations of computing and control in both classical and quantum cases furthermore we show that bounded error tsd can be achieved by a suitable choice of machine precision

the dirichlet problem for the convex envelope the convex envelope of a given function was recently characterized as the solution of a fully nonlinear partial differential equation pde in this article we study a modified problem the dirichlet problem for the underlying pde the main result is an optimal regularity result differentiability c1 regularity of the boundary data implies the corresponding result for the solution in the interior despite the fact that the solution need not be continuous up to the boundary secondary results are the characterization of the convex envelope as i the value function of a stochastic control problem and ii the optimal underestimator for a class of nonlinear elliptic pdes

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

epidemic processes with immunization we study a model of directed percolation dp with immunization i e with different probabilities for the first infection and subsequent infections the immunization effect leads to an additional non markovian term in the corresponding field theoretical action we consider immunization as a small perturbation around the dp fixed point in d 6 where the non markovian term is relevant the immunization causes the system to be driven away from the neighborhood of the dp critical point in order to investigate the dynamical critical behavior of the model we consider the limits of low and high first infection rate while the second infection rate remains constant at the dp critical value scaling arguments are applied to obtain an expression for the survival probability in both limits the corresponding exponents are written in terms of the critical exponents for ordinary dp and dp with a wall we find that the survival probability does not obey a power law behavior decaying instead as a stretched exponential in the low first infection probability limit and to a constant in the high first infection probability limit the theoretical predictions are confirmed by optimized numerical simulations in i 1 dimensions

renormalization of stochastic differential equations with multiplicative noise using effective potential methods we present a method to renormalize stochastic differential equations subjected to multiplicative noise the method is based on the widely used concept of effective potential in high energy physics and has already been successfully applied to the renormalization of stochastic differential equations subjected to additive noise we derive a general formula for the one loop effective potential of a single ordinary stochastic differential equation with arbitrary interaction terms subjected to multiplicative gaussian noise provided the noise satisfies a certain normalization condition to illustrate the usefulness and limitations of the method we use the effective potential to renormalize a toy chemical model based on a simplified gray scott reaction in particular we use it to compute the scale dependence of the toy model s parameters in perturbation theory when subjected to a gaussian power law noise with short time correlations

the obstacle problem for hessian equations on riemannian manifolds abstract in this paper we consider the obstacle problem for a class of fully nonlinear equations on riemannian manifolds the c 1 1 regularity of the greatest viscosity solutions is established

an obstacle problem for a class of monge ampere type functionals abstract in this paper we study an obstacle problem for monge ampere type functionals whose euler lagrange equations are a class of fourth order equations including the affine maximal surface equation and abreu s equation

phase structure of systems with infinite numbers of absorbing states critical properties of systems exhibiting phase transitions into phases with infinite numbers of absorbing states are studied we analyze a non markovian langevin equation recently proposed to describe the critical behavior of such systems and also introduce and study a non markovian discrete model which is argued to present the same critical features on the basis of mean field analysis monte carlo simulations and theoretical arguments we conclude that the phenomenology of the non markovian models closely parallels that of systems with many absorbing states in one and two dimensions the bulk or static critical properties of these systems fall in the directed percolation dp universality class by contrast the critical properties associated with the spread of an initially localized seed exhibit a more complex behavior depending on parameter values they can both in one and two dimensions fall either in the dynamical percolation or dp universality class or else exhibit apparently nonuniversal exponents in contrast to previous results however the nonuniversal exponents in 2d are found to satisfy a scaling law which implies that a particular linear combination of them is universal and assumes dp values these results demonstrate the efficacy of the non markovian approach for understanding systems with many absorbing states which are difficult to analyze in their original microscopic formulation

comment on critical behavior of a two species reaction diffusion problem in a recent paper de freitas et al phys rev e 61 6330 2000 presented simulational results for the critical exponents of the two species reaction diffusion system a stackrel ensuremath rightarrow b 2b and stackrel ensuremath rightarrow b a in dimension d 1 in particular the correlation length exponent was found as ensuremath nu 2 21 5 in contradiction to the well known relation ensuremath nu 2 d in this comment the symmetry arguments leading to exact critical exponents for the universality class to which this reaction diffusion system belongs are concisely reconsidered

learning integer lattices the problem of learning an integer lattice of bf z k in an on line fashion is considered that is the learning algorithm is given a sequence of k tuples of integers and predicts for each tuple in the sequence whether it lies in a hidden target lattice of bf z k the goal of the algorithm is to minimize the number of prediction mistakes an efficient learning algorithm with an absolute mistake bound of k lfloor k log n sqrt k rfloor is given where n is the maximum component of any tuple seen it is shown that this bound is approximately a log log n factor larger than the lower bound on the worst case number of mistakes given by the vc dimension of lattices that are restricted to n cdots 0 cdots n k this algorithm is used to learn rational lattices cosets of lattices an on line word problem for abelian groups and a subclass of the commutative regular languages furthermore by adapting the results of d helmbold r sloan and m k warmuth machine learning

fourth order symplectic integration abstract in this paper we present an explicit fourth order method for the integration of hamilton s equations this method preserves the property that the time evolution of such a system yields a canonical transformation from the initial conditions to the final state that is the integration step is an explicit symplectic map although the result is first derived for a specific type of hamiltonian it is shown to be quite general in particular the results can be applied to any lie group

on learning ring sum expansions the problem of learning ring sum expansions from examples is studied ring sum expansions rse are representations of boolean functions over the base wedge oplus 1 which reflect arithmetic operations in gf 2 k rse is the class of ring sum expansions containing only monomials of length at most k k term is the class of ring sum expansions having at most k monomials it is shown that k rse k geq 1 is learnable while k term rse k geq 2 is not learnable if rp ne np without using a complexity theoretical hypothesis it is proven that k rse k geq 1 and k term rse k geq 2 cannot be learned from positive negative examples alone however if the restriction that the hypothesis which is output by the learning algorithm is also a k rse is suspended then k rse is learnable from positive negative examples only moreover it is proved that 2 term is learnable by a conjunction of a 2 cnf and a 1 dnf finally the paper presents learning on line prediction algorithms for k

some existence results for fully nonlinear elliptic equations of monge ampere type we study the existence and uniqueness of solutions of monge ampere type equations this type of equations has been studied extensively by caffarelli nirenberg spruck and many others see 5 through 8 and the references therein we present some existence and uniqueness results for this type of equations on compact riemannian manifolds with non negative sectional curvature we have also generalized some results in 7

convergence of general decompositions of exponential operators a general theorem is proved concerning the convergence of decompositions of exponential operators in a banach space or normed space as a corollary the convergence of fractal decompositions is proved the convergence of generalized trotter like formulas is also shown to result from the general theorem

a baker campbell hausdorff solution by differential equation we propose a procedure to figure out the baker campbell hausdorff bch solution ln ex ey when the exponent is a linear combination of the spin operator along a direction and its ladder operators the procedure converts the manipulation of the bch formula into that of a differential equation it is shown that the fixed point of the differential equation leads to the solution we are looking for we also remark that the validity of the present method is restricted to the case when the solution branch can be determined in the complex plane

boundary preserving semianalytic numerical algorithms for stochastic differential equations construction of splitting step methods and properties of related nonnegativity and boundary preserving semianalytic numerical algorithms for solving stochastic differential equations sdes of ito type are discussed as the crucial assumption we oppose conditions such that one can decompose the original system of sdes into subsystems for which one knows either the exact solution or its conditional transition probability we present convergence proofs for a newly designed splitting step algorithm and simulation studies for numerous well known numerical examples ranging from stochastic dynamics occurring in asset pricing in mathematical finance cox ingersoll ross cir and constant elasticity of variance cev models to measure valued diffusion and super brownian motion stochastic pdes spdes as met in biology and physics

learning counting functions with queries abstract we investigate the problem of learning disjunctions of counting functions which are general cases of parity and modulo functions with equivalence and membership queries we prove that for any prime number p the class of disjunctions of integer weighted counting functions with modulus p over the domain zqn or zn for any given integer q 2 is polynomial time learnable using at most n 1 equivalence queries where the hypotheses issued by the learner are disjunctions of at most n counting functions with weights from zp in general a counting function may have a composite modulus we prove that for any given integer q 2 over the domain z2n the class of read once disjunctions of boolean weighted counting functions with modulus q is polynomial time learnable with only one equivalence query and o nq membership queries

closed weingarten hypersurfaces in semi riemannian manifolds the existence of closed hypersurfaces of prescribed curvature in semi riemannian manifolds is proved provided there are barriers

general theory of fractal path integrals with applications to many body theories and statistical physics a general scheme of fractal decomposition of exponential operators is presented in any order m namely exp x a b sm x o xm 1 for any positive integer m where sm x et1a et2b et3a et4b etma with finite m depending on m a general recursive scheme of construction of tj is given explicitly it is proven that some of tj should be negative for m 3 and for any finite m nonexistence theorem of positive decomposition general systematic decomposition criterions based on a new type of time ordering are also formulated the decomposition exp x a b sm x n n o xm 1 nm yields a new efficient approach to quantum monte carlo simulations

wilson renormalization of a reaction diffusion process abstract healthy and sick individuals a and b particles diffuse independently with diffusion constants da and db sick individuals upon encounter infect healthy ones at rate k but may also spontaneously recover at rate 1 the propagation of the epidemic therefore couples to the fluctuations in the total population density global extinction occurs below a critical value c of the spatially averaged total density the epidemic evolves as the diffusion reaction decay process a b 2b b a for which we write down the field theory the stationary state properties of this theory when da db were obtained by kree et al the critical behavior for da db remains unsolved

cusps self organization and absorbing states elastic interfaces embedded in quenched random media exhibit metastability and stick slip dynamics these nontrivial dynamical features have been shown to be associated with cusp singularities of the coarse grained disorder correlator here we show that annealed systems with many absorbing states and a conservation law but no quenched disorder exhibit identical cusps on the other hand similar nonconserved systems in the directed percolation class are also shown to exhibit cusps but of a different type these results are obtained both by a recent method to explicitly measure disorder correlators and by defining an alternative new protocol inspired by self organized criticality which opens the door to easily accessible experimental realizations

sticky grains do not change the universality class of isotropic sandpiles we revisit the sandpile model with sticky grains introduced by mohanty and dhar phys rev lett 89 104303 2002 whose scaling properties were claimed to be generically in the universality class of directed percolation for both isotropic and directed models while for directed models this conclusion is unquestionable for isotropic models we present strong evidence that the asymptotic scaling in the self organized regime in which a stationary critical state exists in the limit of slow driving and vanishing dissipation rate is like other stochastic sandpiles generically in the manna universality class this conclusion is drawn from extensive monte carlo simulations and is strengthened by the analysis of the langevin equations proposed by the same authors to account for this problem argued to converge upon coarse graining to the well established set of langevin equations for the manna class

finite size scaling analysis of the critical behavior of a general epidemic process in 2d abstract we investigate the critical behavior of a stochastic lattice model describing a general epidemic process by means of a monte carlo procedure we simulate the model on a regular square lattice and follow the spreading of an epidemic process with immunization a finite size scaling analysis is employed to determine the critical point as well as some critical exponents we show that the usual scaling analysis of the order parameter moment ratio does not provide an accurate estimate of the critical point precise estimates of the critical quantities are obtained from data of the order parameter variation rate and its fluctuations our numerical results corroborate that this model belongs to the dynamic isotropic percolation universality class we also check the validity of the hyperscaling relation and present data collapse curves which reinforce the accuracy of the estimated critical parameters

convergence of unitary and complex decompositions of exponential operators the convergence of the fractal unitary and complex decompositions of exponential operators is proved for hermitian operators in a banach space

numerical study of a field theory for directed percolation a numerical method is devised for the study of stochastic partial differential equations describing directed percolation the contact process and other models with a continuous transition to an absorbing state owing to the heightened sensitivity to fluctuations attending multiplicative noise in the vicinity of an absorbing state a useful method requires discretization of the field variable as well as of space and time when applied to the field theory for directed percolation in 1 1 dimensions the method yields critical exponents which compare well against accepted values

goldberg s theorem and the baker campbell hausdorff formula abstract goldberg s theorem duke math j 23 1956 13 is applied to expressing the logarithm of exponential product in terms of free lie elements using hyperoperators namely inner derivations a symmetric property of the expansion coefficients in the formal power series is also discussed

simulation of spatial systems with demographic noise the demographic shot noise in population dynamics scales with the square root of the population size this process is very important as it yields an absorbing state at zero field but simulating it especially on spatial domains is a nontrivial task here we analyze two similar methods that were suggested for simulating the corresponding langevin equation one by pechenik and levine and the other by dornic chate and munoz dcm these methods are based on operator splitting techniques and the essential difference between them lies in which terms are bundled together in the splitting process both these methods are first order in the time step so one may expect that their performance will be similar we find surprisingly that when simulating the stochastic ginzburg landau equation with two deterministic metastable states the dcm method exhibits two anomalous behaviors first the stochastic stall point moves away from its deterministic counterpart the maxwell point when decreasing the noise second the errors induced by the finite time step are larger by a significant factor i e 10 in the dcm method we show that both these behaviors are the result of a finite time step induced shift in the deterministic maxwell point in the dcm method due to the particular operator splitting employed in light of these results care must be exercised when computing quantities like phase transition boundaries as opposed to universal quantities such as critical exponents in such stochastic spatial systems

reaction diffusion processes and non perturbative renormalization group this paper is devoted to investigating non equilibrium phase transitions to an absorbing state which are generically encountered in reaction diffusion processes it is a review based on canet delamotte deloubriere and wschebor 2004 phys rev lett 92 195703 canet chate and delamotte 2004 phys rev lett 92 255703 and canet et al 2005 phys rev lett 95 100601 of recent progress in this field that has been allowed by a non perturbative renormalization group approach we mainly focus on branching and annihilating random walks and show that their critical properties strongly rely on non perturbative features and that hence the use of a non perturbative method turns out to be crucial to get a correct picture of the physics of these models

interfacial velocity corrections due to multiplicative noise the problem of velocity selection for reaction fronts has been intensively investigated leading to the successful marginal stability ms approach for propagation into an unstable state because the front velocity is controlled by the leading edge which perforce has low density it is interesting to study the role that finite particle number fluctuations have on this picture here we use the well known mapping of discrete markov processes to stochastic differential equations and focus on the front velocity in the simple a a ensuremath leftrightarrows a system our results are consistent with a recent heuristic proposal that v mathrm ms ensuremath v ensuremath sim 1 mathrm ln 2 n

eigenvalue problem of the schrodinger equation via the finite difference time domain method we present a very efficient scheme to calculate the eigenvalue problem of the time independent schrodinger equation the eigenvalue problem can be solved via an initial value procedure of the time dependent schrodinger equation first the time evolution of the wave function is calculated by the finite difference time domain method then the eigenenergies of the electron system can be obtained through a fast fourier transformation along the time axis of the wave function after some point the computing effort for this scheme is roughly proportional to the total grid points involved in the structure and it is suitable for large scale quantum systems we have applied this approach to the three dimensional gan quantum dot system involving one million grid points it takes only 7 h to calculate the confined energies and the wave functions on a standard 2 ghz pentium 4 computer the proposed approach can be implemented in a parallel computer system to study more complex systems

quantum monte carlo methods and general decomposition theory of exponential operators and symplectic integrators abstract the present paper reviews briefly the basic ideas of the quantum monte carlo methods the general decomposition theory of ordered exponentials and symplectic integrators

critical exponents for the restricted sandpile i report large scale monte carlo studies of a one dimensional height restricted stochastic sandpile using the quasistationary simulation method results for systems of up to 50 000 sites yield estimates for critical exponents that differ significantly from those obtained using smaller systems but are consistent with recent predictions derived from a langevin equation for stochastic sandpiles ramasco phys rev e 69 045105 r 2004 this suggests that apparent violations of universality in one dimensional sandpiles are due to strong corrections to scaling and finite size effects