fast numerical computation of 2d free surface jet flow with surface tension a system of partial differential equations that approximate the governing equations for inviscid free surface flow subject to surface tension is presented the approximation is based on linearizing the velocity together with a small scale approximation of the perturbation of the velocity two dirichlet problems must be solved to form the approximate system after which it can be evolved without solving dirichlet problems the accuracy of the solution is determined by how often the velocity term is linearized this time interval is called t we show that the error in the solution of the approximate system at a fixed timetis of the order o t2 we demonstrate numerically that the error is closely correlated to the size of the normal velocity and that there is a stability limit of the form t c un whereundenotes the normal velocity of the free surface and 2 6 importantly cis independent of the resolution so the time step tcan be chosen independently of the number of grid points n this is in contrast to the original system where the stability limit of the time step is t o n 3 2 and a fixed number of dirichlet problems have to be solved per time step by numerical experiments we demonstrate that the approximate system requires less than 10 of the cpu time used by the original system to solve the problem very accurately

playing the hypothesis testing minority game in the maximal reduced strategy space abstract hypothesis testing minority game hmg is a variant of the standard minority game mg that models the inertial behavior of agents in the market in the earlier study of our group we find that agents cooperate better in hmg than in the standard mg when strategies are picked from the full strategy space here we continue to study the behavior of hmg when strategies are chosen from the maximal reduced strategy space surprisingly we find that unlike the standard mg the level of cooperation in hmg depends strongly on the strategy space used in addition a novel intermittency dynamics is also observed in the minority choice time series in a certain parameter range in which the orderly phases are characterized by a variety of periodic dynamics remarkably all these findings can be explained by the crowd anticrowd theory

exact solution of a reaction diffusion process with three site interactions the one dimensional reaction diffusion process aa a and a a aaa is exactly solvable through the empty interval method if the diffusion rate equals the coagulation rate independently of the particle production rate the model is always in the universality class of diffusion annihilation this allows us to check analytically the universality of finite size scaling in a non equilibrium critical point

hopf algebraic structures of suq h cross to 0 2 and suq h cross 2 algebras monopoles and symplectic geometry on 2d manifolds from the symplectic structure on the q deformed manifold sq2 of s2 the classical and quantum q deformed algebras suq h cross to 0 2 and suq h cross 2 of su 2 are discussed in detail and their hopf structures are constructed it is also shown that the symplectic geometry of s2 describes the properties of monopoles for the su 2 algebra the q deformed monopoles are studied from the symplectic geometry of sq2 and a relation between generic 2d manifold and algebraic structures is established

continuous time dynamics of the thermal minority game we study the continuous time dynamics of the thermal minority game we find that the dynamical equations of the model reduce to a set of stochastic differential equations for an interacting disordered system with nontrivial random diffusion this is the simplest microscopic description which accounts for all the features of the system within this framework we study the phase structure of the model and find that its macroscopic properties strongly depend on the initial conditions

on the minority game analytical and numerical studies abstract we investigate further several properties of the minority game we have recently introduced we explain the origin of the phase transition and give an analytical expression of 2 n in the n 2 m region the ability of the players to learn a given payoff is also analyzed and we show that the darwinian evolution process tends to a self organized state in particular the lifetime distribution is a power law with exponent 2 furthermore we study the influence of identical players on their gain and on the systems performance finally we show that large brains always take advantage of small brains

reaction diffusion processes of hard core particles we study a 12 parameter stochastic process involving particles with two site interaction and hard core repulsion on ad dimensional lattice in this model which includes the asymmetric exclusion process contact processes and other processes the stochastic variables are particle occupation numbers taking valuesn x 0 1 we show that on a ten parameter submanifold thek point equal time correlation functions n x1 n xk satisfy linear differential difference equations involving no higher correlators in particular the average density n x satisfies an integrable diffusion type equation these properties are explained in terms of dual processes and various duality relations are derived by defining the time evolution of the stochastic process in terms of a quantum hamiltonianh the model becomes equivalent to a lattice model in thermal equilibrium ind 1 dimensions we show that the spectrum ofh is identical to the spectrum of the quantum hamiltonian of ad dimensional anisotropic spin 1 2 heisenberg model in one dimension our results hint at some new algebraic structure behind the integrability of the system

multichoice minority game dynamics and global cooperation abstract in the original two choice minority game mg selfish players cooperate with each other even though direct communication is not allowed moreover there is a periodic dynamics in the mg whenever the strategy space size is much smaller than the number of strategies at play do these phenomena persist if every player has n c 2 choices where all player s strategies are picked from a reduced strategy space we answer this question by studying a multichoice minority game model known as mg n c s numerical simulation shows that these two models have very similar global cooperative behaviors nevertheless unlike in the mg periodic dynamics does not always appear in the mg n c s even when the strategy space size is much smaller than the number of strategies at play

crowd anticrowd theory of the minority game abstract the minority game is a simple yet highly non trivial agent based model for a complex adaptive system here we provide an explanation of the game s fluctuations which is both intuitive and quantitative and which applies over the entire parameter range of interest the physical idea behind our theory is to describe the interplay between crowds of like minded agents and their anticorrelated partners anticrowds

the exponential accuracy of fourier and chebyshev differencing methods it is shown that when differencing analytic functions using the pseudospectral fourier or chebyshev methods the error committed decays to zero at an exponential rate

stability analysis of finite difference pseudospectral and fourier galerkin approximations for time dependent problems we consider finite difference pseudospectral and fourier galerkin methods for the approximate solution of time dependent problems the paper provides a unified framework for the stability analysis of all three discrete methods in particular the problem of stability for highly accurate stencils is studied in some detail

sparse low energy decomposition for viscous conservation laws for viscous conservation laws solutions contain smooth but high contrast features which require the use of fine grids to properly resolve on coarse grids these high contrast jumps resemble shocks rather than their true viscous profiles which could lead to issues in the numerical approximation of their underlying dynamics in many cases the equations of motion emit traveling wave solutions which can be used to represent the viscous profiles analytically the traveling wave solutions can be thought of as a lower dimensional representation of the motion since they contain information from the evolution equation but are constant along certain time space curves using a parameterized basis involving the traveling waves along with the sparse low energy decompositions found in imaging sciences we propose an approximation to viscous conservation laws which separates the coarse smooth component from the sharp fine one our method provides an appropriate approximation to the solution on a coarse grid thereby accurately under resolving the viscous profile this is similar to the philosophy of shock capturing methods in the sense that we want to capture the viscous front without needing to resolve the profile theoretical results on the consistency of our method are shown in general we provide several computational examples for convex and non convex fluxes

finite element methods for linear hyperbolic problems abstract we give a survey of some recent work by the authors on finite element methods for convectiondiffusion problems and first order linear hyperbolic problems

integrable stochastic ladder models a general way to construct ladder models with certain lie algebraic or quantum lie algebraic symmetries is presented these symmetric models give rise to a series of integrable systems it is shown that corresponding to these su 2 symmetric integrable ladder models there are exactly solvable stationary discrete time resp continuous time markov processes with transition matrices resp intensity matrices having spectra which coincide with the ones of the corresponding integrable models

a spectral lagrange galerkin method for convection dominated diffusion problems abstract the spectral lagrange galerkin method is a numerical technique for time dependent convection diffusion problems based on combining a lagrangian formulation of the equations with the spectral method the resulting scheme can be shown to be unconditionally stable for linear advection diffusion equations and optimal convergence rates may be obtained both the spectral accuracy and the efficiency of the scheme are examined theoretically and by various numerical experiments and extensions of the method to give high accuracy in time are presented

stability and spectral convergence of fourier method for nonlinear problems on the shortcomings of the 2 3 2 3 de aliasing method the high order accuracy of fourier method makes it the method of choice in many large scale simulations we discuss here the stability of fourier method for nonlinear evolution problems focusing on the two prototypical cases of the inviscid burgers equation and the multi dimensional incompressible euler equations the fourier method for such problems with quadratic nonlinearities comes in two main flavors one is the spectral fourier method the other is the 2 3 2 3 pseudo spectral fourier method where one removes the highest 1 3 1 3 portion of the spectrum this is often the method of choice to maintain the balance of quadratic energy and avoid aliasing errors two main themes are discussed in this paper first we prove that as long as the underlying exact solution has a minimal c 1 alpha c 1 spatial regularity then both the spectral and the 2 3 2 3 pseudo spectral fourier methods are stable consequently we prove their spectral convergence for smooth solutions of the inviscid burgers equation and the incompressible euler equations on the other hand we prove that after a critical time at which the underlying solution lacks sufficient smoothness then both the spectral and the 2 3 2 3 pseudo spectral fourier methods exhibit nonlinear instabilities which are realized through spurious oscillations in particular after shock formation in inviscid burgers equation the total variation of bounded pseudo spectral fourier solutions must increase with the number of increasing modes and we stipulate the analogous situation occurs with the 3d incompressible euler equations the limiting fourier solution is shown to enforce l 2 l 2 energy conservation and the contrast with energy dissipating onsager solutions is reflected through spurious oscillations

error estimates for spectral and pseudospectral approximations of hyperbolic equations the linear one dimensional advection equation with variable coefficients and nonperiodic boundary conditions is considered we investigate some spectral and pseudospectral collocation approximations of jacobi and legendre type in the latter case the boundary condition is implicitly treated stability and convergence results in weighted norms are given some precise estimates of the rate of convergence in terms of the regularity of the continuous solution are derived

incorporating inertia into multiagent systems we consider a model that demonstrates the crucial role of inertia and stickiness in multiagent systems based on the minority game the inertia of an agent is introduced into the game model by allowing agents to apply hypothesis testing when choosing their best strategies thereby reducing their reactivity toward changes in the environment we find by extensive numerical simulations that our game shows a remarkable improvement of global cooperation throughout the whole phase space in other words the maladaptation behavior due to over reaction of agents is removed these agents are also shown to be advantageous over the standard ones which are sometimes too sensitive to attain a fair success rate we also calculate analytically the minimum amount of inertia needed to achieve the above improvement our calculation is consistent with the numerical simulation results finally we review some related works in the field that show similar behaviors and compare them to our work

multiple choice minority game abstract minority game is a model of heterogeneous players who think inductively in this game each player chooses one out of two alternatives every turn and those who end up in the minority side wins it is instructive to extend the minority game by allowing players to choose one out of many alternatives nevertheless such an extension is not straight forward due to difficulties in finding a set of reasonable unbiased and computationally feasible strategies here we propose a variation of the minority game where every player has more than two options results of numerical simulations agree with the expectation that our multiple choices minority game exhibits similar behavior as the original two choice minority game

integrable poisson algebras and two dimensional manifolds the relations between integrable poisson algebras with three generators and two dimensional symplectic manifolds are investigated it is shown that for a given integrable poisson algebra there exists a two dimensional symplectic manifold such that the poisson algebra generated by the coordinates of m coincides with the algebra vice versa the coordinates of a given smooth two dimensional symplectic manifold m embedded in generate an integrable poisson algebra moreover smooth poisson algebraic maps between two integrable poisson algebras are governed by equations involving the symplectic manifolds corresponding to these algebras

equivalences between stochastic systems time dependent correlation functions of unstable particles undergoing biased or unbiased diffusion coagulation and annihilation are studied this is achieved by similarity transformations between different stochastic models and between stochastic and soluble nonstochastic models for special cases we obtain exact results which are in good agreement with experiments on one dimensional annihilation coagulation processes

a spectral element method for fluid dynamics laminar flow in a channel expansion a spectral element method that combines the generality of the finite element method with the accuracy of spectral techniques is proposed for the numerical solution of the incompressible navier stokes equations in the spectral element discretization the computational domain is broken into a series of elements and the velocity in each element is represented as a high order lagrangian interpolant through chebyshev collocation points the hyperbolic piece of the governing equations is then treated with an explicit collocation scheme while the pressure and viscous contributions are treated implicitly with a projection operator derived from a variational principle the implementation of the technique is demonstrated on a one dimensional inflow outflow advection diffusion equation and the method is then applied to laminar two dimensional separated flow in a channel expansion comparisons are made with experiment and previous numerical work

removing the stiffness from interfacial flows with surface tension a new formulation and new methods are presented for computing the motion of fluid interfaces with surface tension in two dimensional irrotational and incompressible fluids through the laplace young condition at the interface surface tension introduces high order terms both nonlinear and nonlocal into the dynamics this leads to severe stability constraints for explicit time integration methods and makes the application of implicit methods difficult this new formulation has all the nice properties for time integration methods that are associated with having a linear constant coefficient highest order term that is using this formulation we give implicit time integration methods that have no high order time step stability constraint associated with surface tension and are explicit in fourier space the approach is based on a boundary integral formulation and applies more generally even to problems beyond the fluid mechanical context here they are applied to computing with high resolution the motion of interfaces in hele shaw flows and the motion of free surfaces in inviscid flows governed by the euler equations one hele shaw computation shows the behavior of an expanding gas bubble over long time as the interface undergoes successive tip splittings and finger competition a second computation shows the formation of a very ramified interface through the interaction of surface tension with an unstable density stratification in euler flows the computation of a vortex sheet shows its roll up through the kelvin helmholtz instability this motion culminates in the late time self intersection of the interface creating trapped bubbles of fluid this is we believe a type of singularity formation previously unobserved for such flows in 2d finally computations of falling plumes in an unstably stratified boussinesq fluid show a very similar behavior

stability of pseudospectral and finite difference methods for variable coefficient problems it is shown that pseudospectral approximation to a special class of variable coefficient one dimensional wave equations is stable and convergent even though the wave speed changes sign within the domain computer experiments indicate similar results are valid for more general problems similarly computer results indicate that the leapfrog finite difference scheme is stable even though the wave speed changes sign within the domain however both schemes can be asymptotically unstable in time when a fixed spatial mesh is used

spectral methods for hyperbolic problems we review the current state of fourier and chebyshev collocation methods for the solution of hyperbolic problems with an eye to basic questions of accuracy and stability of the numerical approximations throughout the discussion we emphasize recent developments in the area such as spectral penalty methods the use of filters the resolution of the gibbs phenomenon and issues related to the solution of nonlinear conservations laws such as conservation and convergence we also include a brief discussion on the formulation of multi domain methods for hyperbolic problems and conclude with a few examples of the application of pseudospectral collocation methods for solving nontrivial systems of conservation laws

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100