the range of united k theory abstract we prove that the united k theory functor is a surjective functor from the category of real simple separable purely infinite c algebras to the category of countable acyclic crt modules as a consequence we show that every complex kirchberg algebra satisfying the universal coefficient theorem is the complexification of a real c algebra

coarse stability and bifurcation analysis using time steppers a reaction diffusion example evolutionary pattern forming partial differential equations pdes are often derived as limiting descriptions of microscopic kinetic theory based models of molecular processes e g reaction and diffusion the pde dynamic behavior can be probed through direct simulation time integration or more systematically through stability bifurcation calculations time stepper based approaches like the recursive projection method shroff g m keller h b 1993 siam j numer anal 30 1099 1120 provide an attractive framework for the latter we demonstrate an adaptation of this approach that allows for a direct effective coarse bifurcation analysis of microscopic kinetic based models this is illustrated through a comparative study of the fitzhugh nagumo pde and of a corresponding lattice boltzmann model

periodic homogenization for hypoelliptic diffusions we study the long time behavior of an ornstein uhlenbeck process under the influence of a periodic drift we prove that under the standard diffusive rescaling the law of the particle position converges weakly to the law of a brownian motion whose covariance can be expressed in terms of the solution of a poisson equation we also derive upper bounds on the convergence rate in several metrics

multilevel monte carlo for stochastic differential equations with small noise we consider the problem of numerically estimating expectations of solutions to stochastic differential equations driven by brownian motions in the commonly occurring small noise regime we consider i standard monte carlo methods combined with numerical discretization algorithms tailored to the small noise setting and ii a multilevel monte carlo method combined with a standard euler maruyama implementation under the assumptions we make on the underlying model the multilevel method combined with euler maruyama is often found to be the most efficient option moreover under a wide range of scalings the multilevel method is found to give the same asymptotic complexity that would arise in the idealized case where we have access to exact samples of the required distribution at a cost of o 1 per sample a key step in our analysis is to analyze the variance between two coupled paths directly as opposed to their l 2 distance careful simulations are provided to illustrate the asymptotic results

homogenization of linear and semilinear second order parabolic pdes with periodic coefficients a probabilistic approach we study the limit of the solution of linear and semilinear second order pdes of parabolic type with rapidly oscillating periodic coefficients singular drift and singular coefficient of the zeroth order term our method of proof is fully probabilistic and builds upon the arguments in earlier work in the linear case we use the feynman kac formula to represent the solution of the parabolic pde and in the semilinear case we use an associated backward stochastic differential equation

parametric estimation of stationary stochastic processes under indirect observability for many natural turbulent dynamic systems observed high dimensional dynamic data can be approximated at slow time scales by a process xt driven by a systems of stochastic differential equations sdes when one tries to estimate the parameters of this unobservable sdes systems there is a clear mismatch between the available data and the sdes dynamics to be parametrized here we formalize this indirect observability framework as follows

fitting sde models to nonlinear kac zwanzig heat bath models we study a class of particle in a heat bath models which are a generalization of the well known kac zwanzig class of models but where the coupling between the distinguished particle and the n heat bath particles is through nonlinear springs the heat bath particles have random initial data drawn from an equilibrium gibbs density the primary objective is to approximate the forces exerted by the heat bath which we do not want to resolve by a stochastic process by means of the central limit theorem for gaussian processes and heuristics based on linear response theory we demonstrate conditions under which it is natural to expect that the trajectories of the distinguished particle can be weakly approximated as n by the solution of a markovian sde the quality of this approximation is verified by numerical calculations with parameters chosen according to the linear response theory alternatively the parameters of the effective equation can be chosen using time series analysis this is done and agreement with linear response theory is shown to be good

support recovery for the drift coefficient of high dimensional diffusions consider the problem of learning the drift coefficient of a p dimensional stochastic differential equation from a sample path of length t we assume that the drift is parametrized by a high dimensional vector and study the support recovery problem when both p and t can tend to infinity in particular we prove a general lower bound on the sample complexity t by using a characterization of mutual information as a time integral of conditional variance due to kadota zakai and ziv for linear stochastic differential equations the drift coefficient is parametrized by a p times p matrix that describes which degrees of freedom interact under the dynamics in this case we analyze a ell 1 regularized least squares estimator and prove an upper bound on t that nearly matches the lower bound on specific classes of sparse matrices

coupled channel scattering and separation of coupled differential equations by generalized darboux transformations general methods for separation of coupled differential equations are applied in coupled channel scattering theory with appropriate asymptotic conditions it is shown that the problems with thresholds cannot be separated finally it is pointed out that separation of the coupled channel s wave problem without thresholds can be realized by the use of generalized darboux transformations

estimation of space dependent diffusions and potential landscapes from non equilibrium data in scientific topics ranging from protein folding to the thermohaline ocean circulation it is useful to model the effective macroscopic dynamics of complex systems as noise driven motion in a potential landscape in this paper we consider the estimation of such models from a collection of short non equilibrium trajectories between two points in phase space we generalize a recently introduced spectral methodology for the estimation of diffusion processes from timeseries so that it can be used for non equilibrium data this methodology makes use of the spectral properties leading eigenvalue eigenfunction pairs of the fokker planck operator associated with the diffusion process it is well suited to infer stochastic differential equations that give effective coarse grained descriptions of multiscale systems the generalization to the non equilibrium situation is illustrated with numerical examples in which potentials and diffusion coefficients are estimated from ensembles of short trajectories

path space variational inference for non equilibrium coarse grained systems in this paper we discuss information theoretic tools for obtaining optimized coarse grained molecular models for both equilibrium and non equilibrium molecular simulations the latter are ubiquitous in physicochemical and biological applications where they are typically associated with coupling mechanisms multi physics and or boundary conditions in general the non equilibrium steady states are not known explicitly as they do not necessarily have a gibbs structure the presented approach can compare microscopic behavior of molecular systems to parametric and non parametric coarse grained models using the relative entropy between distributions on the path space and setting up a corresponding path space variational inference problem the methods can become entirely data driven when the microscopic dynamics are replaced with corresponding correlated data in the form of time series furthermore we present connections and generalizations of force matching methods in coarse graining with path space information methods we demonstrate the enhanced transferability of information based parameterizations to different observables at a specific thermodynamic point due to information inequalities we discuss methodological connections between information based coarse graining of molecular systems and variational inference methods primarily developed in the machine learning community however we note that the work presented here addresses variational inference for correlated time series due to the focus on dynamics the applicability of the proposed methods is demonstrated on high dimensional stochastic processes given by overdamped and driven langevin dynamics of interacting particles

an optimization principle for deriving nonequilibrium statistical models of hamiltonian dynamics a general method for deriving closed reduced models of hamiltonian dynamical systems is developed using techniques from optimization and statistical estimation given a vector of resolved variables selected to describe the macroscopic state of the system a family of quasi equilibrium probability densities on phase space corresponding to the resolved variables is employed as a statistical model and the evolution of the mean resolved vector is estimated by optimizing over paths of these densities specifically a cost function is constructed to quantify the lack of fit to the microscopic dynamics of any feasible path of densities from the statistical model it is an ensemble averaged weighted squared norm of the residual that results from submitting the path of densities to the liouville equation the path that minimizes the time integral of the cost function determines the best fit evolution of the mean resolved vector the closed reduced equations satisfied by the optimal path are derived by hamilton jacobi theory when expressed in terms of the macroscopic variables these equations have the generic structure of governing equations for nonequilibrium thermodynamics in particular the value function for the optimization principle coincides with the dissipation potential that defines the relation between thermodynamic forces and fluxes the adjustable closure parameters in the best fit reduced equations depend explicitly on the arbitrary weights that enter into the lack of fit cost function two particular model reductions are outlined to illustrate the general method in each example the set of weights in the optimization principle contracts into a single effective closure parameter

a centre manifold description of containment dispersion in channels with varying flow properties high order asymptotic approximations to the equation governing the longitudinal dispersion of a passive contaminant in poiseuille channel flow are derived and their validity discussed the derivation uses centre manifold theory which provides a systematic and near rigorous approach to calculating each successive approximation it also enables the derivation of the correct initial conditions for the taylor model of shear dispersion approximations that are valid when the channel is of a varying cross section are systematically derived via this approach and the effects of time dependent flow and variable diffusivity are also investigated the resultant modifications to the effective advection velocity and the effective dispersion coefficient are calculated and some general trends indicated

analysis of multiscale methods for stochastic differential equations we analyze a class of numerical schemes proposed in 26 for stochastic differential equations with multiple time scales both advective and diffusive time scales are considered weak as well as strong convergence theorems are proven most of our results are optimal they in turn allow us to provide a thorough discussion on the efficiency as well as optimal strategy for the method c 2005 wiley periodicals inc

discrete approach to stochastic parametrization and dimension reduction in nonlinear dynamics many physical systems are described by nonlinear differential equations that are too complicated to solve in full a natural way to proceed is to divide the variables into those that are of direct interest and those that are not formulate solvable approximate equations for the variables of greater interest and use data and statistical methods to account for the impact of the other variables in the present paper we consider time dependent problems and introduce a fully discrete solution method which simplifies both the analysis of the data and the numerical algorithms the resulting time series are identified by a narmax nonlinear autoregression moving average with exogenous input representation familiar from engineering practice the connections with the mori zwanzig formalism of statistical physics are discussed as well as an application to the lorenz 96 system

concrete construction and properties of the difference equation derived from the cellular automaton using the filtration technique following the proposal of a filtration technique by nobe satsuma and tokihiro we concretely construct partial difference equations which preserve any time evolution patterns of cellular automaton ca stably by the filtration technique we illustrate how to develop a method of filtration for applying to the typical two spatial dimensional ca rule the game of life and verify that the filtration method provides the stable difference equation associated with the ca compared with the inverse ultradiscretization besides in order to discuss whether the filtration technique can lead one to partial differential equations from ca rules we show a derivation of the burgers equation from rule 184 ca via the discrete burgers equation constructed by the filtration method as an example

new stochastic mode reduction strategy for dissipative systems we present a new methodology for studying non hamiltonian nonlinear systems based on an information theoretical extension of a renormalization group technique using a modified maximum entropy principle we obtain a rigorous dimensionally reduced description for such systems the neglected degrees of freedom by this reduction are replaced by a systematically defined stochastic process under a constraint on the second moment this then forms the basis of a computationally efficient method numerical computations for the generalized kuramoto sivashinsky equation support our method and reveal that the long time underlying stochastic process of the fast unresolved modes obeys a universal distribution that does not depend on the initial conditions and which we rigorously derive by the maximum entropy principle

new methods for the two dimensional schrodinger equation susy separation of variables and shape invariance two new methods for the investigation of two dimensional quantum systems whose hamiltonians are not amenable to separation of variables are proposed the first one susy separation of variables is based on the intertwining relations of higher order susy quantum mechanics hsusy qm with supercharges allowing separation of variables the second one is a generalization of shape invariance while in one dimension shape invariance allows us to solve algebraically a class of exactly solvable quantum problems its generalization to higher dimensions has not been explored yet here we provide a formal framework in hsusy qm for two dimensional quantum mechanical systems for which shape invariance holds given the knowledge of one eigenvalue and eigenfunction shape invariance allows us to construct a chain of new eigenfunctions and eigenvalues these methods are applied to a two dimensional quantum system and partial explicit solvability is achieved in the sense that only part of the spectrum is found analytically and a limited set of eigenfunctions is constructed explicitly

solution of second order supersymmetrical intertwining relations in minkowski plane supersymmetrical susy intertwining relations are generalized to the case of quantum hamiltonians in minkowski space for intertwining operators supercharges of second order in derivatives the intertwined hamiltonians correspond to completely integrable systems with the symmetry operators of fourth order in momenta in terms of components the intertwining relations correspond to the system of nonlinear differential equations which are solvable with the simplest constant ansatzes for the metric matrix in second order part of the supercharges the corresponding potentials are built explicitly both for diagonalizable and nondiagonalizable form of metric matrices and their properties are discussed

spectral design for matrix hamiltonians different methods of constructing of a matrix intertwining operator we study relations describing the intertwining of n n matrix in general non hermitian one dimensional hamiltonians by matrix linear differential operators with nondegenerate coefficients of in the highest degree some methods for constructing matrix intertwining operators of the first order of general form are proposed and their interrelations are examined we construct as an example a 2 2 matrix hamiltonian of general form intertwined by an operator of the first order with the hamiltonian with zero matrix potential it is shown that one can add for the final 2 2 matrix hamiltonian with respect to the initial matrix hamiltonian with the help of an intertwining operator of first order up to two bound states for different energy values or up to two bound states described by vector eigenfunctions for the same energy value or up to two bound states described by a vector eigenfunction and an associated vector function for the same energy value

numerical strategies for filtering partially observed stiff stochastic differential equations in this paper we present a fast numerical strategy for filtering stochastic differential equations with multiscale features this method is designed such that it does not violate the practical linear observability condition and more importantly it does not require the computationally expensive cross correlation statistics between multiscale variables that are typically needed in standard filtering approach the proposed filtering algorithm comprises of a macro filter that borrows ideas from the heterogeneous multiscale methods and a micro filter that reinitializes the fast microscopic variables to statistically reflect the unbiased slow macroscopic estimate obtained from the macro filter and macroscopic observations at asynchronous times we will show that the proposed micro filter is equivalent to solving an inverse problem for parameterizing differential equations numerically we will show that this microscopic reinitialization is an important novel feature for accurate filtered solutions especially when the microscopic dynamics is not mixing at all

adaptive sub sampling for parametric estimation of gaussian diffusions we consider a gaussian diffusion xt ornstein uhlenbeck process with drift coefficient and diffusion coefficient 2 and an approximating process y varepsilon t converging to xt in l2 as e 0 we study estimators hat gamma varepsilon hat sigma 2 varepsilon which are asymptotically equivalent to the maximum likelihood estimators of and 2 respectively we assume that the estimators are based on the available n n e observations extracted by sub sampling only from the approximating process y varepsilon t with time step e we characterize all such adaptive sub sampling schemes for which hat gamma varepsilon hat sigma 2 varepsilon are consistent and asymptotically efficient estimators of and 2 as e 0 the favorable adaptive sub sampling schemes are identified by the conditions e 0 0 e and n which implies that we sample from the process y varepsilon t with a vanishing but coarse time step e e this study highlights the necessity to sub sample at adequate rates when the observations are not generated by the underlying stochastic model whose parameters are being estimated the adequate sub sampling rates we identify seem to retain their validity in much wider contexts such as the additive triad application we briefly outline

the role of additive and multiplicative noise in filtering complex dynamical systems covariance inflation is an ad hoc treatment that is widely used in practical real time data assimilation algorithms to mitigate covariance underestimation owing to model errors nonlinearity or an

pauli equation and the method of supersymmetric factorization we consider different variants of factorization of a 2 2 matrix schrodinger pauli operator in two spatial dimensions they allow us to relate its spectrum to the sum of spectra of two scalar schrodinger operators in a manner similar to one dimensional darboux transformations we consider both the case when such factorization is reduced to the ordinary two dimensional supersymmetric quantum mechanics quasifactorization and a more general case which involves covariant derivatives the admissible classes of electromagnetic fields are described and some illustrative examples are given

higher order matrix susy transformations in two dimensional quantum mechanics the iteration procedure of supersymmetric transformations for the two dimensional schrodinger operator is implemented by means of the matrix form of factorization in terms of matrix 2 2 supercharges two different types of iterations are investigated in detail the particular case of diagonal initial hamiltonian is considered and the existence of solutions is demonstrated explicit examples illustrate the construction

a new framework for extracting coarse grained models from time series with multiscale structure in many applications it is desirable to infer coarse grained models from observational data the observed process often corresponds only to a few selected degrees of freedom of a high dimensional dynamical system with multiple time scales in this work we consider the inference problem of identifying an appropriate coarse grained model from a single time series of a multiscale system it is known that estimators such as the maximum likelihood estimator or the quadratic variation of the path estimator can be strongly biased in this setting here we present a novel parametric inference methodology for problems with linear parameter dependency that does not suffer from this drawback furthermore we demonstrate through a wide spectrum of examples that our methodology can be used to derive appropriate coarse grained models from time series of partial observations of a multiscale system in an effective and systematic fashion

a tale of two time scales it is a common practice in finance to estimate volatility from the sum of frequently sampled squared returns however market microstructure poses challenges to this estimation approach as evidenced by recent empirical studies in finance the present work attempts to lay out theoretical grounds that reconcile continuous time modeling and discrete time samples we propose an estimation approach that takes advantage of the rich sources in tick by tick data while preserving the continuous time assumption on the underlying returns under our framework it becomes clear why and where the usual volatility estimator fails when the returns are sampled at the highest frequencies if the noise is asymptotically small our work provides a way of finding the optimal sampling frequency a better approach the two scales estimator works for any size of the noise

a priori tests of a stochastic mode reduction strategy several a priori tests of a systematic stochastic mode reduction procedure recently devised by the authors proc natl acad sci 96 1999 14687 commun pure appl math 54 2001 891 are developed here in this procedure reduced stochastic equations for a smaller collections of resolved variables are derived systematically for complex nonlinear systems with many degrees of freedom and a large collection of unresolved variables while the above approach is mathematically rigorous in the limit when the ratio of correlation times between the resolved and the unresolved variables is arbitrary small it is shown here on a systematic hierarchy of models that this ratio can be surprisingly big typically the systematic reduced stochastic modeling yields quantitatively realistic dynamics for ratios as large as 1 2 the examples studied here vary from instructive stochastic triad models to prototype complex systems with many degrees of freedom utilizing the truncated burgers hopf equations as a nonlinear heat bath systematic quantitative tests for the stochastic modeling procedure are developed here which involve the stationary distribution and the two time correlations for the second and fourth moments including the resolved variables and the energy in the resolved variables in an important illustrative example presented here the nonlinear original system involves 102 degrees of freedom and the reduced stochastic model predicted by the theory for two resolved variables involves both nonlinear interaction and multiplicative noises even for large value of the correlation time ratio of the order of 1 2 the reduced stochastic model with two degrees of freedom captures the essentially nonlinear and non gaussian statistics of the original nonlinear systems with 102 modes extremely well furthermore it is shown here that the standard regression fitting of the second order correlations alone fails to reproduce the nonlinear stochastic dynamics in this example 2002 elsevier science b v all rights reserved

nonlinearity and temporal dependence nonlinearities in the drift and diffusion coefficients influence temporal dependence in diffusion models we study this link using three measures of temporal dependence mixing mixing and mixing stationary diffusions that are mixing have mixing coefficients that decay exponentially to zero when they fail to be mixing they are still mixing and mixing but coefficient decay is slower than exponential for such processes we find transformations of the markov states that have finite variances but infinite spectral densities at frequency zero the resulting spectral densities behave like those of stochastic processes with long memory finally we show how state dependent poisson sampling alters the temporal dependence

approximation of complex symmetric operators an operator t on a complex hilbert space mathcal h is called a complex symmetric operator if there exists a conjugate linear isometric involution c mathcal h rightarrow mathcal h so that ctc t in this paper we study the approximation of complex symmetric operators by virtue of an intensive analysis of compact operators in singly generated c algebras we obtain a complete characterization of norm limits of complex symmetric operators and provide a classification of complex symmetric operators up to approximate unitary equivalence this gives a general solution to the norm closure problem for complex symmetric operators as an application we provide a concrete description of partial isometries which are norm limits of complex symmetric operators

parametric estimation from approximate data non gaussian diffusions we study the problem of parameters estimation in indirect observability contexts where x t in r r is an unobservable stationary process parametrized by a vector of unknown parameters and all observable data are generated by an approximating process y varepsilon t which is close to x t in l 4 norm we construct consistent parameter estimators which are smooth functions of the sub sampled empirical mean and empirical lagged covariance matrices computed from the observable data we derive explicit optimal sub sampling schemes specifying the best paired choices of sub sampling time step and number of observations we show that these choices ensure that our parameter estimators reach optimized asymptotic l 2 convergence rates which are constant multiples of the l 4 norm y varepsilon t x t

parameter estimation for multiscale diffusions we study the problem of parameter estimation for time series possessing two widely separated characteristic time scales the aim is to understand situations where it is desirable to fit a homogenized single scale model to such multiscale data we demonstrate numerically and analytically that if the data is sampled too finely then the parameter fit will fail in that the correct parameters in the homogenized model are not identified we also show numerically and analytically that if the data is subsampled at an appropriate rate then it is possible to estimate the coefficients of the homogenized model correctly the ideas are studied in the context of thermally activated motion in a two scale potential however the ideas may be expected to transfer to other situations where it is desirable to fit an averaged or homogenized equation to multiscale data

an ensemble kalman filter for statistical estimation of physics constrained nonlinear regression models a central issue in contemporary science is the development of nonlinear data driven statistical dynamical models for time series of noisy partial observations from nature or a complex model it has been established recently that ad hoc quadratic multi level regression models can have finite time blow up of statistical solutions and or pathological behavior of their invariant measure recently a new class of physics constrained nonlinear regression models were developed to ameliorate this pathological behavior here a new finite ensemble kalman filtering algorithm is developed for estimating the state the linear and nonlinear model coefficients the model and the observation noise covariances from available partial noisy observations of the state several stringent tests and applications of the method are developed here in the most complex application the perfect model has 57 degrees of freedom involving a zonal east west jet two topographic rossby waves and 54 nonlinearly interacting rossby waves the perfect model has significant non gaussian statistics in the zonal jet with blocked and unblocked regimes and a non gaussian skewed distribution due to interaction with the other 56 modes we only observe the zonal jet contaminated by noise and apply the ensemble filter algorithm for estimation numerically we find that a three dimensional nonlinear stochastic model with one level of memory mimics the statistical effect of the other 56 modes on the zonal jet in an accurate fashion including the skew non gaussian distribution and autocorrelation decay on the other hand a similar stochastic model with zero memory levels fails to capture the crucial non gaussian behavior of the zonal jet from the perfect 57 mode model