comparison of systems with complex behavior abstract we present a formalism for comparing the asymptotic dynamics of dynamical systems with physical systems that they model based on the spectral properties of the koopman operator we first compare invariant measures and discuss this in terms of a statistical takens theorem proved here we also identify the need to go beyond comparing only invariant ergodic measures of systems and introduce an ergodic theoretic treatment of a class of spectral functionals that allow for this the formalism is extended for a class of stochastic systems discrete random dynamical systems the ideas introduced in this paper can be used for parameter identification and model validation of driven nonlinear models with complicated behavior as an illustration we provide an example in which we compare the asymptotic behavior of a combustion system measured experimentally with the asymptotic behavior of a class of models that have the form of a random dynamical system

semiparametric forecasting and filtering correcting low dimensional model error in parametric models semiparametric forecasting and filtering are introduced as a method of addressing model errors arising from unresolved physical phenomena while traditional parametric models are able to learn high dimensional systems from small data sets their rigid parametric structure makes them vulnerable to model error on the other hand nonparametric models have a very flexible structure but they suffer from the curse of dimensionality and are not practical for high dimensional systems the semiparametric approach loosens the structure of a parametric model by fitting a data driven nonparametric model for the parameters given a parametric dynamical model and a noisy data set of historical observations an adaptive kalman filter is used to extract a time series of the parameter values a nonparametric forecasting model for the parameters is built by projecting the discrete shift map onto a data driven basis of smooth functions existing techniques for filtering and forecasting algorithms extend naturally to the semiparametric model which can effectively compensate for model error with forecasting skill approaching that of the perfect model semiparametric forecasting and filtering are a generalization of statistical semiparametric models to time dependent distributions evolving under dynamical systems

some quantum lie algebras of type dn positive a quantum lie algebra is constructed within the positive part of the drinfeld jimbo quantum group of type dn our quantum lie algebra structure includes a generalized antisymmetry property and a generalized jacobi identity closely related to the braid equation a generalized universal enveloping algebra of our quantum lie algebra of type dn positive is proved to be the drinfeld jimbo quantum group of the same type the existence of such a generalized lie algebra is reduced to an integer programming problem moreover when the integer programming problem is feasible we show by means of the generalized jacobi identity that the poincare birkhoff witt theorem basis is still true

finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple lie algebra let g be a complex simple lie algebra we show that whent is not a root of 1 all finite dimensional representations of the quantum analogu t g are completely reducible and we classify the irreducible ones in terms of highest weights in particular they can be seen as deformations of the representations of the classical u g

optimal filtering of complex turbulent systems with memory depth through consistency constraints in this article we develop a linear theory for optimal filtering of complex turbulent signals with model errors through linear autoregressive models we will show that when the autoregressive model parameters are chosen such that they satisfy absolute stability and consistency conditions of at least order 2 of the classical multistep method for solving initial value problems the filtered solutions with autoregressive models of order p 2 are optimal in the sense that they are comparable to the estimates obtained from the true filter with perfect model this result is reminiscent of the lax equivalence fundamental theorem in the analysis of finite difference discretization scheme for the numerical solutions of partial differential equations we will apply this linear theory to filter two nonlinear problems the slowest mode of the truncated burgers hopf and the lorenz 96 model on these nonlinear problems we will show that whenever these linear conditions are satisfied the filtered solutions accuracy is significantly improved finally we will also apply the recently developed offline test criteria to understand the robustness of the multistep filter on various turbulent nature including the stochastically forced linear advection diffusion equation and a toy model for barotropic turbulent rossby waves

classification of bicovariant differential calculi on quantum groups suppose thatq is not a root of unity we classify all bicovariant differential calculi of dimension greater than one on the quantum groupsgl q n o q n andsp q n for which the differentials du of the matrix entriesu generate the left module of first order forms our first classification theorem asserts that there are precisely two one parameter families of such calculi ongl q n forn 3 in the limitq 1 only two of these calculi give the ordinary differential calculus ongl n our second main theorem states that apart from finitely manyq there exist precisely two differential calculi with these properties ono q n andsp q n forn 4 this strengthens the corresponding result proved in our previous paper ss2 there are four such calculi ono q 3 we introduce two new 4 dimensional bicovariant differential calculi ono q 3

matrix elements and wigner coefficients for uq gl n in this series of papers the fundamental wigner coefficients and the matrix elements of the generators for the quantum group uq gl n are derived the uq gl n uq gl n 1 reduced wigner coefficients and reduced matrix elements are determined algebraically as eigenvalues of certain uq gl n 1 invariants in the quantum group uq gl n the matrix elements of the elementary quantum group generators in the gel fand tsetlin basis are derived below while all fundamental wigner coefficients are derived in the second paper of the series

stochastic mode reduction for the immersed boundary method we apply the formulation of a stochastic mode reduction method developed in a recent paper of majda timofeyev and vanden eijnden comm pure appl math 54 2001 pp 891 974 mtv to obtain simplified equations for the dynamics of structures immersed in a thermally fluctuating fluid at low reynolds or kubo number as simulated by a recent extension of the immersed boundary ib method by kramer and peskin proceedings of the second mit conference on computational fluid and solid mechanics elsevier science oxford uk 2003 pp 1755 1758 the effective dynamics of the immersed structures are not obvious in the primitive equations which involve both fluid and structure dynamics but the procedure of mtv allows the rigorous derivation of a reduced stochastic system for the immersed structures alone we find in the limit of small reynolds or kubo number that the lagrangian particle constituents of the immersed structures undergo a drift diffusive motion with several physically correct featur

the nonuniform hard rod fluid revisited the statistical mechanics of the one dimensional nonuniform pure hard core fluid is formulated in the spirit of the reiss frisch lebowitz rfl scaled particle theory by emphasizing the importance of the core dependence a more intuitive and simpler derivation can be given the wiener hopf type construction of the pair direct correlation function is formulated via the dyson variational method of inverse scattering theory which is compared with the particle hole theory the new approach allows us to lift the global free energy functional into a larger space where all the symmetries become apparent

principal dynamical components a procedure is proposed for a dimension reduction in time series similarly to principal components the procedure seeks a low dimensional manifold that minimizes information loss unlike principal components however the procedure involves dynamical considerations through the proposal of a predictive dynamical model in the reduced manifold hence the minimization of the uncertainty is not only over the choice of a reduced manifold as in principal components but also over the parameters of the dynamical model as in autoregressive analysis and principal interaction patterns further generalizations are provided to nonautonomous and non markovian scenarios which are then applied to historical sea surface temperature data 2012 wiley periodicals inc

blended reduced subspace algorithms for uncertainty quantification of quadratic systems with a stable mean state abstract order reduction schemes have been used successfully for the analysis and simplification of high dimensional systems exhibiting low dimensional dynamics in this work we first focus on presenting generic limitations of order reduction techniques in systems with stable mean state that exhibit irreducible high dimensional features such as non normal dynamics wide energy spectra or strong energy cascades between modes the reduced order framework that we consider to illustrate these limitations is the dynamically orthogonal do field equations this framework is applied to a series of examples with stable mean state including a linear non normal system and a nonlinear triad system in various dynamical configurations after illustrating the weaknesses and generic limitations of order reduction we develop a novel two way coupled blended approach based on the quasilinear gaussian qg closure and the do field equations the new method qg do overcomes the limitations of its two ingredients and achieves exceptional performance in the examples described previously as well as in other configurations with strongly transient character without using any tuned or adjustable parameters

dynamic stochastic superresolution of sparsely observed turbulent systems real time capture of the relevant features of the unresolved turbulent dynamics of complex natural systems from sparse noisy observations and imperfect models is a notoriously difficult problem the resulting lack of observational resolution and statistical accuracy in estimating the important turbulent processes which intermittently send significant energy to the large scale fluctuations hinders efficient parameterization and real time prediction using discretized pde models this issue is particularly subtle and important when dealing with turbulent geophysical systems with an vast range of interacting spatio temporal scales and rough energy spectra near the mesh scale of numerical models here we introduce and study a suite of general dynamic stochastic superresolution dss algorithms and show that by appropriately filtering sparse regular observations with the help of cheap stochastic exactly solvable models one can derive stochastically superresolved velocity fields and gain insight into the important characteristics of the unresolved dynamics including the detection of the so called black swans the dss algorithms operate in fourier domain and exploit the fact that the coarse observation network aliases high wavenumber information into the resolved waveband it is shown that these cheap algorithms are robust and have significant skill on a test bed of turbulent solutions from realistic nonlinear turbulent spatially extended systems in the presence of a significant model error in particular the dss algorithms are capable of successfully capturing time localized extreme events in the unresolved modes and they provide good and robust skill for recovery of the unresolved processes in terms of pattern correlation moreover we show that dss improves the skill for recovering the primary modes associated with the sparse observation mesh which is equally important in applications the skill of the various dss algorithms depends on the energy spectrum of the turbulent signal and the observation time relative to the decorrelation time of the turbulence at a given spatial scale in a fashion elucidated here the dss technique exploiting a simple gaussian closure of the nonlinear stochastic forecast model emerges as the most suitable trade off between the superresolution skill and computational complexity associated with estimating the cross correlations between the aliasing modes of the sparsely observed turbulent signal such techniques offer a promising and efficient approach to constraining unresolved turbulent fluxes through stochastic superparameterization and a subsequent improvement in coarse grained filtering and prediction of the next generation atmosphere ocean system aos models

assimilating irregularly spaced sparsely observed turbulent signals with hierarchical bayesian reduced stochastic filters in this paper we consider a practical filtering approach for assimilating irregularly spaced sparsely observed turbulent signals through a hierarchical bayesian reduced stochastic filtering framework the proposed hierarchical bayesian approach consists of two steps blending a data driven interpolation scheme and the mean stochastic model msm filter we examine the potential of using the deterministic piecewise linear interpolation scheme and the ordinary kriging scheme in interpolating irregularly spaced raw data to regularly spaced processed data and the importance of dynamical constraint through msm in filtering the processed data on a numerically stiff state estimation problem in particular we test this approach on a two layer quasi geostrophic model in a two dimensional domain with a small radius of deformation to mimic ocean turbulence our numerical results suggest that the dynamical constraint becomes important when the observation noise variance is large second we find that the filtered estimates with ordinary kriging are superior to those with linear interpolation when observation networks are not too sparse such robust results are found from numerical simulations with many randomly simulated irregularly spaced observation networks various observation time intervals and observation error variances third when the observation network is very sparse we find that both the kriging and linear interpolations are comparable

classification of bicovariant differential calculi on the jordanian quantum groups and and quantum lie algebras assuming that the bicovariant bimodules are generated as left modules by the differentials of the quantum group generators we classify all four dimensional first order bicovariant calculi on the jordanian quantum group and all three dimensional first order bicovariant calculi on the jordanian quantum group it is found that there are three one parameter families of four dimensional bicovariant first order calculi on and that there is a single unique three dimensional bicovariant calculus on this three dimensional calculus may be obtained through a classical like reduction from any one of the three families of four dimensional calculi on details of the higher order calculi and also the quantum lie algebras are presented for all calculi the quantum lie algebra obtained from the bicovariant calculus on is shown to be isomorphic to the quantum lie algebra we obtain as an ad submodule within the jordanian universal enveloping algebra and also through a consideration of the decomposition of the tensor product of two copies of the deformed adjoint module we also obtain the quantum killing form for this quantum lie algebra

models for stochastic climate prediction there has been a recent burst of activity in the atmosphere ocean sciences community in utilizing stable linear langevin stochastic models for the unresolved degree of freedom in stochastic climate prediction here several idealized models for stochastic climate modeling are introduced and analyzed through unambiguous mathematical theory this analysis demonstrates the potential need for more sophisticated models beyond stable linear langevin equations the new phenomena include the emergence of both unstable linear langevin stochastic models for the climate mean and the need to incorporate both suitable nonlinear effects and multiplicative noise in stochastic models under appropriate circumstances the strategy for stochastic climate modeling that emerges from this analysis is illustrated on an idealized example involving truncated barotropic flow on a beta plane with topography and a mean flow in this example the effect of the original 57 degrees of freedom is well represented by a theoretically predicted stochastic model with only 3 degrees of freedom

differential calculus on compact matrix pseudogroups quantum groups the paper deals with non commutative differential geometry the general theory of differential calculus on quantum groups is developed bicovariant bimodules as objects analogous to tensor bundles over lie groups are studied tensor algebra and external algebra constructions are described it is shown that any bicovariant first order differential calculus admits a natural lifting to the external algebra so the external derivative of higher order differential forms is well defined and obeys the usual properties the proper form of the cartan maurer formula is found the vector space dual to the space of left invariant differential forms is endowed with a bilinear operation playing the role of the lie bracket commutator generalized antisymmetry relation and jacobi identity are proved

a comparative study of two stochastic mode reduction methods abstract we present a comparative study of two methods for the reduction of the dimensionality of a system of ordinary differential equations that exhibits time scale separation both methods lead to a reduced system of stochastic differential equations the novel feature of these methods is that they allow the use in the reduced system of higher order terms in the resolved variables the first method proposed by majda timofeyev and vanden eijnden is based on an asymptotic strategy developed by kurtz the second method is a short memory approximation of the mori zwanzig projection formalism of irreversible statistical mechanics as proposed by chorin hald and kupferman we present conditions under which the reduced models arising from the two methods should have similar predictive ability we apply the two methods to test cases that satisfy these conditions the form of the reduced models and the numerical simulations show that the two methods have similar predictive ability as expected

a computational strategy for multiscale systems with applications to lorenz 96 model numerical schemes for systems with multiple spatio temporal scales are investigated the multiscale schemes use asymptotic results for this type of systems which guarantee the existence of an effective dynamics for some suitably defined modes varying slowly on the largest scales the multiscale schemes are analyzed in general then illustrated on a specific example of a moderately large deterministic system displaying chaotic behavior due to lorenz issues like consistency accuracy and efficiency are discussed in detail the role of possible hidden slow variables as well as additional effects arising on the diffusive time scale are also investigated as a byproduct we obtain a rather complete characterization of the effective dynamics in lorenz model

dynamical criteria for the evolution of the stochastic dimensionality in flows with uncertainty abstract we estimate and study the evolution of the dominant dimensionality of dynamical systems with uncertainty governed by stochastic partial differential equations within the context of dynamically orthogonal do field equations transient nonlinear dynamics irregular data and non stationary statistics are typical in a large range of applications such as oceanic and atmospheric flow estimation to efficiently quantify uncertainties in such systems it is essential to vary the dimensionality of the stochastic subspace with time an objective here is to provide criteria to do so working directly with the original equations of the dynamical system under study and its do representation we first analyze the scaling of the computational cost of these do equations with the stochastic dimensionality and show that unlike many other stochastic methods the do equations do not suffer from the curse of dimensionality subsequently we present the new adaptive criteria for the variation of the stochastic dimensionality based on instantaneous i stability arguments and ii bayesian data updates we then illustrate the capabilities of the derived criteria to resolve the transient dynamics of two 2d stochastic fluid flows specifically a double gyre wind driven circulation and a lid driven cavity flow in a basin in these two applications we focus on the growth of uncertainty due to internal instabilities in deterministic flows we consider a range of flow conditions described by varied reynolds numbers and we study and compare the evolution of the uncertainty estimates under these varied conditions

quantum lie algebras their existence uniqueness and q antisymmetry quantum lie algebras are generalizations of lie algebras which have the quantum parameter h built into their structure they have been defined concretely as certain submodules of the quantized enveloping algebras on them the quantum lie product is given by the quantum adjoint action

a mathematical framework for stochastic climate models there has been a recent burst of activity in the atmosphere ocean sciences community in utilizing stable linear langevin stochastic models for the unresolved degrees of freedom in stochastic climate prediction here a systematic mathematical strategy for stochastic climate modeling is developed and some of the new phenomena in the resulting equations for the climate variables alone are explored the new phenomena include the emergence of both unstable linear langevin stochastic models for the climate mean variables and the need to incorporate both suitable nonlinear effects and multiplicative noise in stochastic models under appropriate circumstances all of these phenomena are derived from a systematic self consistent mathematical framework for eliminating the unresolved stochastic modes that is mathematically rigorous in a suitable asymptotic limit the theory is illustrated for general quadratically nonlinear equations where the explicit nature of the stochastic climate modeling procedure can be elucidated the feasibility of the approach is demonstrated for the truncated equations for barotropic flow with topography explicit concrete examples with the new phenomena are presented for the stochastically forced three mode interaction equations the conjecture of smith and waleffe phys fluids 11 1999 1608 1622 for stochastically forced three wave resonant equations in a suitable regime of damping and forcing is solved as a byproduct of the approach examples of idealized climate models arising from the highly inhomogeneous equilibrium statistical mechanics for geophysical flows are also utilized to demonstrate self consistency of the mathematical approach with the predictions of equilibrium statistical mechanics in particular for these examples the reduced stochastic modeling procedure for the climate variables alone is designed to reproduce both the climate mean and the energy spectrum of the climate variables c 2001 john wiley sons inc

filtering nonlinear spatio temporal chaos with autoregressive linear stochastic models abstract fundamental barriers in practical filtering of nonlinear spatio temporal chaotic systems are model errors attributed to the stiffness in resolving multiscale features recently reduced stochastic filters based on linear stochastic models have been introduced to overcome such stiffness one of them is the mean stochastic model msm based on a diagonal ornstein uhlenbeck process in fourier space despite model errors the msm shows very encouraging filtering skill especially when the hidden signal of interest is strongly chaotic in this regime the dynamical system statistical properties resemble to those of the energy conserving equilibrium statistical mechanics with gaussian invariant measure therefore the ornstein uhlenbeck process with appropriate parameters is sufficient to produce reasonable statistical estimates for the filter model in this paper we consider a generalization of the msm with a diagonal autoregressive linear stochastic model in fourier space as a filter model for chaotic signals with long memory depth with this generalization the filter prior model becomes slightly more expensive than the msm but it is still less expensive relative to integrating the perfect model which is typically unknown in real problems furthermore the associated kalman filter on each fourier mode is computationally as cheap as inverting a matrix of size d where d is the number of observed variables on each fourier mode in our numerical example d 1 using the lorenz 96 l 96 model as a testbed we show that the non markovian nature of this autoregressive model is an important feature in capturing the highly oscillatory modes with long memory depth second we show that the filtering skill with autoregressive models supersedes that with msm in weakly chaotic regime where the memory depth is longer in strongly chaotic regime the performance of the ar p filter is still better or at least comparable to that of the msm most importantly we find that this reduced filtering strategy is not as sensitive as standard ensemble filtering strategies to additional intrinsic model errors that are often encountered when model parameters are incorrectly specified

fluctuation properties of steady state langevin systems motivated by stochastic models of climate phenomena the steady state of a linear stochastic model with additive gaussian white noise is studied fluctuation theorems for nonequilibrium steady states provide a constraint on the character of these fluctuations the properties of the fluctuations which are unconstrained by the fluctuation theorem are investigated and related to the model parameters the irreversibility of trajectory segments which satisfies a fluctuation theorem is used as a measure of nonequilibrium fluctuations the moments of the irreversibility probability density function pdf are found and the pdf is seen to be non gaussian the average irreversibility goes to zero for short and long trajectory segments and has a maximum for some finite segment length which defines a characteristic time scale of the fluctuations the initial average irreversibility growth rate is equal to the average entropy production and is related to noise amplification for systems with a separation of deterministic time scales modes with time scales much shorter than the trajectory time span and whose noise amplitudes are not asymptotically large do not to first order contribute to the irreversibility statistics providing a potential basis for dimensional reduction

classification of bicovariant differential calculi on quantum groups of type a b c and d under the assumptions thatq is not a root of unity and that the differentialsduji of the matrix entries span the left module of first order forms we classify bicovariant differential calculi on quantum groupsan 1 bn cn anddn we prove that apart one dimensional differential calculi and from finitely many values ofq there are precisely2n such calculi on the quantum groupan 1 slq n forn 3 all these calculi have the dimensionn2 for the quantum groupsbn cn anddn we show that except for finitely manyq there exist precisely twon2 dimensional bicovariant calculi forn 3 wheren 2n 1 forbn andn 2n forcn dn the structure of these calculi is explicitly described and the corresponding ad invariant right ideals of ker e are determined in the limitq 1 two of the 2n calculi foran 1 and one of the two calculi forbn cn anddn contain the ordinary classical differential calculus on the corresponding lie group as a quotient

quantum lie algebras associated to and quantum lie algebras are non associative algebras which are embedded into the quantized enveloping algebras of drinfeld and jimbo in the same way as ordinary lie algebras are embedded into their enveloping algebras the quantum lie product on is induced by the quantum adjoint action of we construct the quantum lie algebras associated to and we determine the structure constants and the quantum root systems which are now functions of the quantum parameter q they exhibit an interesting duality symmetry under

stochastic homogenization for an energy conserving multi scale toy model of the atmosphere we study a hamiltonian toy model for a lagrangian fluid parcel in the semi geostrophic limit which exhibits slow and fast dynamics we first reinject unresolved fast dynamics into the deterministic equation through a stochastic parametrization that respects the conservation of the energy of the deterministic system in a second step we use stochastic singular perturbation theory to derive an effective reduced stochastic differential equation for the slow dynamics we verify the results in numerical simulations

quantum deformations of certain simple modules over enveloping algebras soit f un corps soit f 0 un sous corps de f soit u l algebre enveloppante universelle sur f 0 de l algebre de lie de kac moody correspondant a la matrice a d entiers on montre qu un module integrable de poids le plus eleve v de u admet une deformation quantique

predictions of critical transitions with non stationary reduced order models abstract here we demonstrate the ability of stochastic reduced order models to predict the statistics of non stationary systems undergoing critical transitions first we show that the reduced order models are able to accurately predict the autocorrelation function and probability density functions pdf of higher dimensional systems with time dependent slow forcing of either the resolved or unresolved modes second we demonstrate that whether the system tips early or repeatedly jumps between the two equilibrium points flickering depends on the strength of the coupling between the resolved and unresolved modes and the time scale separation both kinds of behaviour have been found to precede critical transitions in earlier studies furthermore we demonstrate that the reduced order models are also able to predict the timing of critical transitions the skill of various proposed tipping indicators are discussed

bicovariant quantum algebras and quantum lie algebras a bicovariant calculus of differential operators on a quantum group is constructed in a natural way using invariant maps from fun mathfrak g q tou q g given by elements of the pure braid group these operators the reflection matrix y l sl being a special case generate algebras that linearly close under adjoint actions i e they form generalized lie algebras we establish the connection between the hopf algebra formulation of the calculus and a formulation in compact matrix form which is quite powerful for actual computations and as applications we find the quantum determinant and an orthogonality relation fory inso q n

a note on the generalized lie algebra in a recent paper dobrev and sudbery classified the highest weight and lowest weight finite dimensional irreducible representations of the quantum lie algebra introduced by lyubashenko and sudbery the aim of this note is to add to this classification all the finite dimensional irreducible representations which have no highest weight and or no lowest weight in the case when q is a root of unity for this purpose we give a description of the enlarged centre

normal forms for reduced stochastic climate models the systematic development of reduced low dimensional stochastic climate models from observations or comprehensive high dimensional climate models is an important topic for atmospheric low frequency variability climate sensitivity and improved extended range forecasting here techniques from applied mathematics are utilized to systematically derive normal forms for reduced stochastic climate models for low frequency variables the use of a few empirical orthogonal functions eofs also known as principal component analysis karhunen loeve and proper orthogonal decomposition depending on observational data to span the low frequency subspace requires the assessment of dyad interactions besides the more familiar triads in the interaction between the low and high frequency subspaces of the dynamics it is shown below that the dyad and multiplicative triad interactions combine with the climatological linear operator interactions to simultaneously produce both strong nonlinear dissipation and correlated additive and multiplicative cam stochastic noise for a single low frequency variable the dyad interactions and climatological linear operator alone produce a normal form with cam noise from advection of the large scales by the small scales and simultaneously strong cubic damping these normal forms should prove useful for developing systematic strategies for the estimation of stochastic models from climate data as an illustrative example the one dimensional normal form is applied below to low frequency patterns such as the north atlantic oscillation nao in a climate model the results here also illustrate the short comings of a recent linear scalar cam noise model proposed elsewhere for low frequency variability

interpolating irregularly spaced observations for filtering turbulent complex systems we present a numerically fast reduced filtering strategy the fourier domain kalman filter with appropriate interpolations to account for irregularly spaced observations of complex turbulent signals the design of such a reduced filter involves i interpolating irregularly spaced observations to the model regularly spaced grid points ii understanding under which situation the small scale oscillatory artifact from such interpolation won t degrade the filtered solutions iii understanding when the interpolated covariance structure can be approximated by its diagonal terms when observations are corrupted by independent gaussian noise and iv applying a scalar kalman filter formula on each fourier component independently with an approximate diagonal interpolated covariance matrix from the practical point of view there is an emerging need to understand the effect of i toward the filtered solutions for example in utilizing the data produced from various interpolation techniques that merge multiple satellite measurements of atmospheric and ocean dynamical quantities to understand point iii above and to see how many of nondiagonal terms are effectively ignored in iv we compute a ratio lambda between the largest nondiagonal components and the smallest diagonal components of the interpolated covariance matrix we find that for piecewise linear interpolation this ratio lambda is always smaller than that of the alternative interpolation schemes such as trigonometric and cubic spline for any irregularly spaced observation networks when observations are not so sparse with small noise we find that the small scale oscillatory artifact in ii above is negligible when piecewise linear interpolation is used whereas for the other schemes such as the nearest neighbor trigonometric and cubic spline interpolation the oscillatory artifact degrades the filtered solutions significantly finally we also find that the reduced filtering strategy with piecewise linear interpolation produces more accurate filtered solutions than conventional approaches when observations are extremely irregularly spaced such that the ratio lambda is not so small and very sparse

stochastic mode reduction for particle based simulation methods for complex microfluid systems we illustrate the stochastic mode reduction procedure as formulated recently by majda timofeyev and vanden eijnden comm pure appl math 54 2001 pp 891 974 mtv on the equations of motion underlying various particle based simulation approaches such as stokesian dynamics and brownian dynamics and the conceptually distinct dissipative particle dynamics dpd simulation approaches for complex microfluid systems the resulting coarse grained dynamics are compared and contrasted with each other we show that the stochastic mode reduction procedure provides a way to recover the smoluchowski dynamics for a standard model of multiple interacting particles in a fluid the dpd however has some subtle aspects which obstruct the application of the stochastic mode reduction procedure we discuss the mathematical and physical properties of the dpd method that underlie thisdifficulty

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

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

filtering skill for turbulent signals for a suite of nonlinear and linear extended kalman filters the filtering skill for turbulent signals from nature is often limited by errors due to utilizing an imperfect forecast model in particular real time filtering and prediction when very limited or no a posteriori analysis is possible e g spread of pollutants storm surges tsunami detection etc introduces a number of additional challenges to the problem here a suite of filters implementing stochastic parameter estimation for mitigating model error through additive and multiplicative bias correction is examined on a nonlinear exactly solvable stochastic test model mimicking turbulent signals in regimes ranging from configurations with strongly intermittent transient instabilities associated with positive finite time lyapunov exponents to laminar behavior stochastic parameterization extended kalman filter spekf used as a benchmark here involves exact formulas for propagating the mean and covariance of the augmented forecast model including the unresolved parameters the remaining filters use the same nonlinear forecast model but they introduce model error through different moment closure approximations and or linear tangent approximation used for computing the second order statistics of the augmented stochastic forecast model a comprehensive study of filter performance is carried out in the presence of various moment closure errors which are enhanced by additional model errors due to incorrect parameters inducing additive and multiplicative stochastic biases the estimation skill of the unresolved stochastic parameters is also discussed and it is shown that the linear tangent filter despite its popularity is completely unreliable in many turbulent regimes for both parameter estimation and filtering moreover regimes of filter divergence for the linear tangent filter are identified the results presented here provide useful guidelines for filtering turbulent high dimensional spatially extended systems with more general model errors as well as for designing more skillful methods for superparameterization of unresolved intermittent processes in complex multi scale models they also provide unambiguous benchmarks for the capabilities of linear and nonlinear extended kalman filters using incorrect statistics on an exactly solvable test bed with rich and realistic dynamics

the structure of quantum lie algebras for the classical series and the structure constants of quantum lie algebras depend on a quantum deformation parameter q and they reduce to the classical structure constants of a lie algebra at q 1 we explain the relationship between the structure constants of quantum lie algebras and quantum clebsch gordan coefficients for we present a practical method for the determination of these quantum clebsch gordan coefficients and are thus able to give explicit expressions for the structure constants of the quantum lie algebras associated to the classical lie algebras and in the quantum case the structure constants of the cartan subalgebra are non zero and we observe that they are determined in terms of the simple quantum roots we introduce an invariant killing form on the quantum lie algebras and find that it takes values which are simple q deformations of the classical ones

efficient statistically accurate algorithms for the fokker planck equation in large dimensions abstract solving the fokker planck equation for high dimensional complex turbulent dynamical systems is an important and practical issue however most traditional methods suffer from the curse of dimensionality and have difficulties in capturing the fat tailed highly intermittent probability density functions pdfs of complex systems in turbulence neuroscience and excitable media in this article efficient statistically accurate algorithms are developed for solving both the transient and the equilibrium solutions of fokker planck equations associated with high dimensional nonlinear turbulent dynamical systems with conditional gaussian structures the algorithms involve a hybrid strategy that requires only a small number of ensembles here a conditional gaussian mixture in a high dimensional subspace via an extremely efficient parametric method is combined with a judicious non parametric gaussian kernel density estimation in the remaining low dimensional subspace particularly the parametric method provides closed analytical formulae for determining the conditional gaussian distributions in the high dimensional subspace and is therefore computationally efficient and accurate the full non gaussian pdf of the system is then given by a gaussian mixture different from traditional particle methods each conditional gaussian distribution here covers a significant portion of the high dimensional pdf therefore a small number of ensembles is sufficient to recover the full pdf which overcomes the curse of dimensionality notably the mixture distribution has significant skill in capturing the transient behavior with fat tails of the high dimensional non gaussian pdfs and this facilitates the algorithms in accurately describing the intermittency and extreme events in complex turbulent systems it is shown in a stringent set of test problems that the method only requires an order of o 100 ensembles to successfully recover the highly non gaussian transient pdfs in up to 6 dimensions with only small errors

numerical approximation of some linear stochastic partial differential equations driven by special additive noises this paper is concerned with the numerical approximation of some linear stochastic partial differential equations with additive noises a special representation of the noise is considered and it is compared with general representations of noises in the infinite dimensional setting convergence analysis and error estimates are presented for the numerical solution based on the standard finite difference and finite element methods the effects of the noises on the accuracy of the approximations are illustrated results of the numerical experiments are provided

predictability of extreme events in a nonlinear stochastic dynamical model the objective of this work is to evaluate the potential of reduced order models to reproduce the extreme event and predictability characteristics of higher dimensional dynamical systems a nonlinear toy model is used which contains key features of comprehensive climate models first we demonstrate that the systematic stochastic mode reduction strategy leads to a reduced order model with the same extreme value characteristics as the full dynamical models for a wide range of time scale separations second we find that extreme events in this model follow a generalized pareto distribution with a negative shape parameter thus extreme events are bounded in this model third we show that a precursor approach has good forecast skill for extreme events we then find that the reduced stochastic models capture the predictive skill of extreme events of the full dynamical models well consistent with previous studies we also find that the larger the extreme events the better predictable they are our results suggest that systematically derived reduced order models have the potential to be used for the modeling and statistical prediction of weather and climate related extreme events and possibly in other areas of science and engineering too

on quantum lie algebras and quantum root systems as a natural generalization of ordinary lie algebras we introduce the concept of quantum lie algebras we define these in terms of certain adjoint submodules of quantized enveloping algebras endowed with a quantum lie bracket given by the quantum adjoint action the structure constants of these algebras depend on the quantum deformation parameter q and they go over into the usual lie algebras when q 1 the notions of q conjugation and q linearity are introduced q linear analogues of the classical antipode and cartan involution are defined and a generalized killing form q linear in the first entry and linear in the second is obtained these structures allow the derivation of symmetries between the structure constants of quantum lie algebras the explicitly worked out examples of and illustrate the results

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

reduced models of atmospheric low frequency variability parameter estimation and comparative performance abstract low frequency variability lfv of the atmosphere refers to its behavior on time scales of 10 100 days longer than the life cycle of a mid latitude cyclone but shorter than a season this behavior is still poorly understood and hard to predict the present study compares various model reduction strategies that help in deriving simplified models of lfv three distinct strategies are applied here to reduce a fairly realistic high dimensional quasi geostrophic 3 level qg3 atmospheric model to lower dimensions i an empirical dynamical method which retains only a few components in the projection of the full qg3 model equations onto a specified basis and finds the linear deterministic and the stochastic corrections empirically as in selten 1995 5 ii a purely dynamics based technique employing the stochastic mode reduction strategy of majda et al 2001 62 and iii a purely empirical multi level regression procedure which specifies the functional form of the reduced model and finds the model coefficients by multiple polynomial regression as in kravtsov et al 2005 3 the empirical dynamical and dynamical reduced models were further improved by sequential parameter estimation and benchmarked against multi level regression models the extended kalman filter was used for the parameter estimation overall the reduced models perform better when more statistical information is used in the model construction thus the purely empirical stochastic models with quadratic nonlinearity and additive noise reproduce very well the linear properties of the full qg3 model s lfv i e its autocorrelations and spectra as well as the nonlinear properties i e the persistent flow regimes that induce non gaussian features in the model s probability density function the empirical dynamical models capture the basic statistical properties of the full model s lfv such as the variance and integral correlation time scales of the leading lfv modes as well as some of the regime behavior features but fail to reproduce the detailed structure of autocorrelations and distort the statistics of the regimes dynamical models that use data assimilation corrections do capture the linear statistics to a degree comparable with that of empirical dynamical models but do much less well on the full qg3 model s nonlinear dynamics these results are discussed in terms of their implications for a better understanding and prediction of lfv

low dimensional chaotic dynamics versus intrinsic stochastic noise a paradigm model several prototype models are introduced here which are designed to elucidate the interaction between heteroclinic low dimensional chaos in the projected nonlinear dynamics and intrinsic stochasticity induced by energy exchange with a bath of fast variables these models are built by coupling a four dimensional ode with known analytical properties including heteroclinic cycles with a suitable deterministic bath of fast variables a systematic strategy for stochastic mode reduction is applied to these models with 104 degrees of freedom to derive four dimensional reduced stochastic equations for the slow variables due to the internal chaotic dynamics of the slow variables the stochastic mode reduction strategy is very robust in this case and yields reduced models which accurately capture the statistical behavior of the original deterministic system furthermore it is also shown here that even in the regime of a weak coupling between the slow variables and the fast heat bath the detailed structure of the stochastic terms derived through the mode elimination procedure is essential for reproducing the statistical behavior of the slow dynamics