an obstruction based approach to the kochen specker theorem in isham and butterfield 1998 int j theor phys 37 2669 733 it was shown that the kochen specker theorem can be written in terms of the non existence of global elements of a certain varying set over the category of boolean subalgebras of projection operators on some hilbert space in this paper we show how obstructions to the construction of such global elements arise and how this provides a new way of looking at proofs of the theorem

resonant response in nonequilibrium steady states the time dependent probability density function of the order parameter of a system evolving toward a stationary state exhibits an oscillatory behavior if the eigenvalues of the corresponding evolution operator are complex the frequencies n with which the system reaches its stationary state correspond to the imaginary part of such eigenvalues if the system at the stationary state is further driven by a small and oscillating perturbation with a given frequency we formally prove that the linear response to the probability density function is enhanced when n for n n we prove that the occurrence of this phenomenon is characteristic of systems that are in a nonequilibrium stationary state in particular we obtain an explicit formula for the frequency dependent mobility in terms of the relaxation to the stationary state of the unperturbed probability current we test all these predictions by means of numerical simulations considering an ensemble of noninteracting overdamped particles on a tilted periodic potential

a wavelet monte carlo method for turbulent diffusion with many spatial scales a new algorithm is developed here for monte carlo simulation of turbulent diffusion with random velocity fields having a power law spatial spectrum arbitrarily many statistical scales long range correlations and infrared divergence this algorithm uses an expansion of the moving average representation of a gaussian random field via a specific orthonormal basis which exploits the scaling behavior of the random field and yields a compact representation of the field despite infrared divergence the orthonormal basis involves a family of wavelets due to alpert and rokhlin and the vanishing of a large number of moments for this basis order 3 or 4 is crucial for compact representation of the statistical velocity field the authors also develop a rigorous practical a priori energy criterion for truncation of the wavelet expansion furthermore for moment cancellations with orders 3 or 4 and a fixed energy tolerance the number of gaussian random basis elements needed by the method is sublinear in the exponent m where 1 x 2m denotes the range of active velocity scales the algorithm developed here has remarkably low variance as regards sampling errors for example for the infrared divergent velocity spectrum corresponding to the kolmogoroff spectrum in an exactly solvable model the velocity structure function is simulated accurately for separation distances over more than five decades with only 792 random basis elements and 100 ensemble realizations with this same singular spectrum in the model problem the pair dispersion statistic for the passive scalar can be determined within errors of 3 throughout over five decades of pair separation distance with only 792 random elements 1000 realizations and a few hours calculation on a workstation all of the computational algorithms and results are presented for a nontrivial exactly solvable model and are described within a mathematical framework of error analysis where analytic sampling and discretization errors are treated separately

on spurious detection of linear response and misuse of the fluctuation dissipation theorem in finite time series abstract using a sensitive statistical test we determine whether or not one can detect the breakdown of linear response given observations of deterministic dynamical systems a goodness of fit statistics is developed for a linear statistical model of the observations based on results for central limit theorems for deterministic dynamical systems and used to detect linear response breakdown we apply the method to discrete maps which do not obey linear response and show that the successful detection of breakdown depends on the length of the time series the magnitude of the perturbation and on the choice of the observable we find that in order to reliably reject the assumption of linear response for typical observables sufficiently large data sets are needed even for simple systems such as the logistic map one needs of the order of 10 6 observations to reliably detect the breakdown with a confidence level of 95 if less observations are available one may be falsely led to conclude that linear response theory is valid the amount of data required is larger the smaller the applied perturbation for judiciously chosen observables the necessary amount of data can be drastically reduced but requires detailed a priori knowledge about the invariant measure which is typically not available for complex dynamical systems furthermore we explore the use of the fluctuation dissipation theorem fdt in cases with limited data length or coarse graining of observations the fdt if applied naively to a system without linear response is shown to be very sensitive to the details of the sampling method resulting in erroneous predictions of the response

random shearing direction models for isotropic turbulent diffusion recently a rigorous renormalization theory for various scalar statistics has been developed for special modes of random advection diffusion involving random shear layer velocity fields with long range spatiotemporal correlations new random shearing direction models for isotropic turbulent diffusion are introduced here in these models the velocity field has the spatial second order statistics of an arbitrary prescribed stationary incompressible isotropic random field including long range spatial correlations with infrared divergence but the temporal correlations have finite range the explicit theory of renormalization for the mean and second order statistics is developed here with e the spectral parameter for e 4 and measuring the strength of the infrared divergence of the spatial spectrum the scalar mean statistics rigorously exhibit a phase transition from mean field behavior for e 2 to anomalous behavior for e with 2 e 4 as conjectured earlier by avellaneda and the author the universal inertial range renormalization for the second order scalar statistics exhibits a phase transition from a covariance with a gaussian functional form for e with e 2 to an explicit family with a non gaussian covariance for e with 2 e 4 these non gaussian distributions have tails that are broader than gaussian as e varies with 2 e 4 and behave for large values like exp c c x 4 e withc c an explicit constant also here the attractive general principle is formulated and proved that every steady stationary zero mean isotropic incompressible gaussian random velocity field is well approximated by a suitable superposition of random shear layers

solitary shock waves and other travelling waves in a general compressible hyperelastic rod in the literature it has been conjectured that solitary shock waves can arise in incompressible hyperelastic rods recently it has been shown that this conjecture is true one might guess that when compressibility is taken into account such a wave which is both a solitary wave and a shock wave can still arise one of the aims of this paper is to show the existence of this interesting type of wave in general compressible hyperelastic rods and provide an analytical description it is difficult to directly tackle the fully nonlinear rod equations here by using a non dimensionalization process and the reductive perturbation technique we derive a new type of nonlinear dispersive equation as the model equation we then focus on the travelling wave solutions of this new equation as a result we obtain a system of ordinary differential equations an important feature of this system is that there is a vertical singular line in the phase plane which leads to the appearance of shock waves by considering the equilibrium points and their relative positions to the singular line we are able to determine all qualitatively different phase planes those paths in phase planes which represent physically acceptable solutions are discussed one by one it turns out that there is a variety of travelling waves including solitary shock waves solitary waves periodic shock waves etc analytical expressions for all these waves are obtained a new phenomenon is also found a solitary wave can suddenly change into a periodic wave with finite period in dynamical systems this represents a homoclinic orbit suddenly changing into a closed orbit to the authors9 knowledge such a bifurcation has not been found in any other dynamical systems

evidence of dispersion relations for the nonlinear response of the lorenz 63 system along the lines of the nonlinear response theory developed by ruelle in a previous paper we have proved under rather general conditions that kramers kronig dispersion relations and sum rules apply for a class of susceptibilities describing at any order of perturbation the response of axiom a non equilibrium steady state systems to weak monochromatic forcings we present here the first evidence of the validity of these integral relations for the linear and the second harmonic response for the perturbed lorenz 63 system by showing that numerical simulations agree up to high degree of accuracy with the theoretical predictions some new theoretical results showing how to derive asymptotic behaviors and how to obtain recursively harmonic generation susceptibilities for general observables are also presented our findings confirm the conceptual validity of the nonlinear response theory suggest that the theory can be extended for more general non equilibrium steady state systems and shed new light on the applicability of very general tools based only upon the principle of causality for diagnosing the behavior of perturbed chaotic systems and reconstructing their output signals in situations where the fluctuation dissipation relation is not of great help

challenges in climate science and contemporary applied mathematics this article discusses the challenges in climate science from the emerging viewpoint of stochastic statistical properties of turbulent dynamical systems the mathematical topics discussed here include empirical information theory fluctuation dissipation theorems reduced order stochastic modeling and the development of mathematically unambiguous exactly solvable test models for climate science that capture crucial features of vastly more complex scientific problems the applied mathematics topics include the emerging development of multiscale algorithms for filtering data assimilation and superparametrization for climate science and other problems in science and engineering as well as suitable unambiguous mathematical test problems for their behavior interesting contemporary research directions and specific open problems are mentioned throughout the article the perspective here should also be useful for applications to other complex dynamical systems arising in neural science material science and environmental mechanical engineering 2011 wiley periodicals inc

existence and uniqueness of weak solutions for precipitation fronts a novel hyperbolic free boundary problem in several space variables the determination of the large scale boundaries between moist and dry regions is an important problem in contemporary meteorology these phenomena have been addressed recently in a simplified tropical climate model through a novel hyperbolic free boundary formulation yielding three families drying slow moistening and fast moistening of precipitation fronts the last two wave types violate lax s shock inequalities yet are robustly realized this formal hyperbolic free boundary problem is given here a rigorous mathematical basis by establishing the existence and uniqueness of suitable weak solutions arising in the zero relaxation limit a new l2 contraction estimate is also established at positive relaxation values 2010 wiley periodicals inc

direct numerical simulation based measurement of the mean impulse response of homogeneous isotropic turbulence a technique for measuring the mean impulse response function of stationary homogeneous isotropic turbulence is proposed such a measurement is carried out here on the basis of direct numerical simulation dns a zero mean white noise volume forcing is used to probe the turbulent flow and the response function is obtained by accumulating the space time correlation between the white forcing and the velocity field this technique to measure the turbulent response in a dns numerical experiment is a research tool in that field of spectral closures where the linear response concept is invoked either by resorting to renormalized perturbations theories or by introducing the well known fluctuation dissipation relation fdr although the results obtained in the present work are limited to relatively low values of the reynolds number a preliminary analysis is possible both the characteristic form and the time scaling properties of the response function are investigated in the universal subrange of dissipative wave numbers a comparison with the response approximation given by the fdr is proposed through the independent dns measurement of the correlation function very good agreement is found between the measured response and kraichnan s description of random energy range advection effects

differentiation of srb states correction and complements taking into account criticism by d dolgopyat and m jiang we present here an improved derivation of the formula for the derivative of an srb measure with respect to parameters

colouring the rational quantum sphere and the kochen specker theorem we review and extend recent findings of godsil and zaks who published a constructive colouring of the rational unit sphere with the property that for any orthogonal tripod formed by rays extending from the origin of the points of the sphere exactly one ray is red one white and one black they also showed that any consistent colouring of the real sphere requires an additional colour we discuss some of the consequences for the kochen specker theorem

response theory for equilibrium and non equilibrium statistical mechanics causality and generalized kramers kronig relations we consider the general response theory recently proposed by ruelle for describing the impact of small perturbations to the non equilibrium steady states resulting from axiom a dynamical systems we show that the causality of the response functions entails the possibility of writing a set of kramers kronig k k relations for the corresponding susceptibilities at all orders of nonlinearity nonetheless only a special class of directly observable susceptibilities obey k k relations specific results are provided for the case of arbitrary order harmonic response which allows for a very comprehensive k k analysis and the establishment of sum rules connecting the asymptotic behavior of the harmonic generation susceptibility to the short time response of the perturbed system these results set in a more general theoretical framework previous findings obtained for optical systems and simple mechanical models and shed light on the very general impact of considering the principle of causality for testing self consistency the described dispersion relations constitute unavoidable benchmarks that any experimental and model generated dataset must obey the theory exposed in the present paper is dual to the time dependent theory of perturbations to equilibrium states and to non equilibrium steady states and has in principle similar range of applicability and limitations in order to connect the equilibrium and the non equilibrium steady state case we show how to rewrite the classical response theory by kubo so that response functions formally identical to those proposed by ruelle apart from the measure involved in the phase space integration are obtained these results taking into account the chaotic hypothesis by gallavotti and cohen might be relevant in several fields including climate research in particular whereas the fluctuation dissipation theorem does not work for non equilibrium systems because of the non equivalence between internal and external fluctuations k k relations might be robust tools for the definition of a self consistent theory of climate change

two simple proofs of the kochen specker theorem a new proof of the kochen specker theorem uses 33 rays instead of 117 in the original proof if the number of dimensions is increased from 3 to 4 only 24 rays are needed

deviations from fluctuation relaxation relations abstract fluctuation relaxation relations connect correlation functions of dynamical variables with the mean decay of amplitudes these relations are revisited attention is given to deviations associated with departure from the canonical distribution examples are taken from fluid turbulence

stable resonances and signal propagation in a chaotic network of coupled units we apply the linear response theory developed by ruelle j stat phys 95 393 1999 to analyze how a periodic signal of weak amplitude superimposed upon a chaotic background is transmitted in a network of nonlinearly interacting units we numerically compute the complex susceptibility and show the existence of specific poles stable resonances corresponding to the response to perturbations transverse to the attractor contrary to the poles of correlation functions they depend on the pair emitting receiving units this dynamic differentiation induced by nonlinearities exhibits the different ability that units have to transmit a signal in this network

smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics this paper reviews various applications of the theory of smooth dynamical systems to conceptual problems of nonequilibrium statistical mecanics we adopt a new point of view which has emerged progressively in recent years and which takes seriously into account the chaotic character of the microscopic time evolution the emphasis is on nonequilibrium steady states rather than the traditional approach to equilibrium point of view of boltzmann the nonequilibrium steady states in presence of a gaussian thermostat are described by srb measures in terms of these one can prove the gallavotti cohen fluctuation theorem one can also prove a general linear response formula and study its consequences which are not restricted to near equilibrium situations at equilibrium one recovers in particular the onsager reciprocity relations under suitable conditions the nonequilibrium steady states satisfy the pairing theorem of dettmann and morriss the results just mentioned hold so far only for classical systems they do not involve large size i e they hold without a thermodynamic limit

transfer of energy in camassa holm and related models by use of nonunique characteristics abstract we study the propagation of energy density in finite energy weak solutions of the camassa holm and related equations developing the methods based on generalized nonunique characteristics we show that the parts of energy related to positive and negative slopes are one sided weakly continuous and of bounded variation which allows us to define certain measures of dissipation of both parts of energy the result is a step towards the open problem of uniqueness of dissipative solutions of the camassa holm equation

improving filtering and prediction of spatially extended turbulent systems with model errors through stochastic parameter estimation the filtering and predictive skill for turbulent signals is often limited by the lack of information about the true dynamics of the system and by our inability to resolve the assumed dynamics with sufficiently high resolution using the current computing power the standard approach is to use a simple yet rich family of constant parameters to account for model errors through parameterization this approach can have significant skill by fitting the parameters to some statistical feature of the true signal however in the context of real time prediction such a strategy performs poorly when intermittent transitions to instability occur alternatively we need a set of dynamic parameters one strategy for estimating parameters on the fly is a stochastic parameter estimation through partial observations of the true signal in this paper we extend our newly developed stochastic parameter estimation strategy the stochastic parameterization extended kalman filter spekf to filtering sparsely observed spatially extended turbulent systems which exhibit abrupt stability transition from time to time despite a stable average behavior for our primary numerical example we consider a turbulent system of externally forced barotropic rossby waves with instability introduced through intermittent negative damping we find high filtering skill of spekf applied to this toy model even in the case of very sparse observations with only 15 out of the 105 grid points observed and with unspecified external forcing and damping additive and multiplicative bias corrections are used to learn the unknown features of the true dynamics from observations we also present a comprehensive study of predictive skill in the one mode context including the robustness toward variation of stochastic parameters imperfect initial conditions and finite ensemble effect furthermore the proposed stochastic parameter estimation scheme applied to the same spatially extended rossby wave system demonstrates high predictive skill comparable with the skill of the perfect model for a duration of many eddy turnover times especially in the unstable regime

exceptional and non crystallographic root systems and the kochen specker theorem the kochen specker theorem states that a 3 dimensional complex euclidean space admits a finite configuration of projective lines such that the corresponding quantum observables the orthogonal projectors cannot be assigned with 0 and 1 values in a classically consistent way this paper shows that the irreducible root systems of exceptional and of non crystallographic types are useful in constructing such configurations in other dimensions the cases e 6 and e 7 lead to new examples while f 4 e 8 and h 4 yield a new interpretation of the known ones the described configurations have an additional property they are saturated i e the tuples of mutually orthogonal lines being partially ordered by inclusion yield a poset with all maximal elements having the same cardinality the dimension of space

multi level dynamical systems connecting the ruelle response theory and the mori zwanzig approach we consider the problem of deriving approximate autonomous dynamics for a number of variables of a dynamical system which are weakly coupled to the remaining variables in a previous paper we have used the ruelle response theory on such a weakly coupled system to construct a surrogate dynamics such that the expectation value of any observable agrees up to second order in the coupling strength to its expectation evaluated on the full dynamics we show here that such surrogate dynamics agree up to second order to an expansion of the mori zwanzig projected dynamics this implies that the parametrizations of unresolved processes suited for prediction and for the representation of long term statistical properties are closely related if one takes into account in addition to the widely adopted stochastic forcing the often neglected memory effects

towards a general theory of extremes for observables of chaotic dynamical systems in this paper we provide a connection between the geometrical properties of the attractor of a chaotic dynamical system and the distribution of extreme values we show that the extremes of so called physical observables are distributed according to the classical generalised pareto distribution and derive explicit expressions for the scaling and the shape parameter in particular we derive that the shape parameter does not depend on the chosen observables but only on the partial dimensions of the invariant measure on the stable unstable and neutral manifolds the shape parameter is negative and is close to zero when high dimensional systems are considered this result agrees with what was derived recently using the generalized extreme value approach combining the results obtained using such physical observables and the properties of the extremes of distance observables it is possible to derive estimates of the partial dimensions of the attractor along the stable and the unstable directions of the flow moreover by writing the shape parameter in terms of moments of the extremes of the considered observable and by using linear response theory we relate the sensitivity to perturbations of the shape parameter to the sensitivity of the moments of the partial dimensions and of the kaplan yorke dimension of the attractor preliminary numerical investigations provide encouraging results on the applicability of the theory presented here the results presented here do not apply for all combinations of axiom a systems and observables but the breakdown seems to be related to very special geometrical configurations

a note on the camassa holm equation abstract we consider a generalized camassa holm equation we improve the compactness argument of 5 6 7 17 establishing the existence of global weak solutions in h 1 without the use of an ole nik type estimate the potential interest of this improvement is linked to convergence analysis of numerical schemes for which it may not be easy to verify an ole nik type estimate

a test model for fluctuation dissipation theorems with time periodic statistics abstract the recently developed time periodic fluctuation dissipation theorem fdt provides a very convenient way of addressing the climate change of atmospheric systems with seasonal cycle by utilizing statistics of the present climate a triad nonlinear stochastic model with exactly solvable first and second order statistics is introduced here as an unambiguous test model for fdt in a time periodic setting this model mimics the nonlinear interaction of two rossby waves forced by baroclinic processes with a zonal jet forced by a polar temperature gradient periodic forcing naturally introduces the seasonal cycle into the model the exactly solvable first and second order statistics are utilized to compute both the ideal mean and variance response to the perturbations in forcing or dissipation and the quasi gaussian approximation of fdt qg fdt that uses the mean and the covariance in the equilibrium state the time averaged mean and variance qg fdt response to perturbations of forcing or dissipation is compared with the corresponding ideal response utilizing the triad test model in a number of regimes with various dynamical and statistical properties such as weak or strong non gaussianity and resonant or non resonant forcing it is shown that even in a strongly non gaussian regime qg fdt has surprisingly high skill for the mean response to the changes in forcing on the other hand the performance of qg fdt for the variance response to the perturbations of dissipation is good in the near gaussian regime and deteriorates in the strongly non gaussian regime the results here on the test model should provide useful guidelines for applying the time periodic fdt to more complex realistic systems such as atmospheric general circulation models

algebraic partial boolean algebras partial boolean algebras first studied by kochen and specker in the 1960s provide the structure for bell kochen specker theorems which deny the existence of non contextual hidden variable theories in this paper we study partial boolean algebras which are algebraic in the sense that their elements have coordinates in an algebraic number field several of these algebras have been discussed recently in a debate on the validity of bell kochen specker theorems in the context of finite precision measurements the main result of this paper is that every algebraic finitely generated partial boolean algebra b t is finite when the underlying space is three dimensional answering a question of kochen and showing that conway and kochen s infinite algebraic partial boolean algebra has minimum dimension this result contrasts the existence of an infinite non algebraic b t generated by eight elements in an abstract orthomodular lattice of height 3 we then initiate a study of higher dimensional algebraic partial boolean algebras first we describe a restriction on the determinants of the elements of b t that are generated by a given set t we then show that when the generating set t consists of the rays spanning the minimal vectors in a real irreducible root lattice b t is infinite just if that root lattice has an a5 sublattice finally we characterize the rays of b t when t consists of the rays spanning the minimal vectors of the root lattice e8

a simple justification of the singular limit for equatorial shallow water dynamics the equatorial shallow water equations at low froude number form a symmetric hyperbolic system with large variable coefficient terms although such systems are not covered by the classical klainerman majda theory of singular limits the first two authors recently proved that solutions exist uniformly and converge to the solutions of the long wave equations as the height and froude number tend to 0 their proof exploits the special structure of the equations by expanding solutions in series of parabolic cylinder functions a simpler proof of a slight generalization is presented here in the spirit of the classical theory 2008 wiley periodicals inc

the relevance of chaos for the linear response theory abstract we discuss the van kampen objection about the range of validity of the linear response theory we show that in chaotic systems the instability of the dynamics has a constructive role for any system with good chaotic properties a fluctuation response relation holds at macroscopic times because the relevant perturbative parameter is t c f where t c is a correlation time and f the strength of the external force nevertheless the van kampen criticism has relevant consequences on the feasibility of numerical experiments

resonances for axiom bf a flows etant donne un flot d axiome a sur m et des fonctions lisses b c m r on montre que la fonction de correlation en temps bc pour un etat de gibbs a une transformee de fourier bc mesomorphe dans une bande on etudie les residus des poles de bc

linear response of the lorenz system the present numerical study provides strong evidence that at standard parameters the response of the lorenz system to small perturbations of the control parameter r is linear this evidence is obtained not only directly by determining the response in the observable a x z but also indirectly by validating various implications of the assumption of a linear response like a quadratic response at twice the perturbation frequency a vanishing response in a x x the kramers kronig relations and relations between different response functions since the lorenz system is nonhyperbolic the present results indicate that in contrast to a recent speculation the large system limit thermodynamic limit need not be invoked to obtain a linear response for chaotic systems of this type

kochen specker theorem for finite precision spin one measurements unsharp spin one observables arise from the fact that a residual uncertainty about the actual orientation of the measurement device remains if the uncertainty is below a certain level and if the distribution of measurement errors is covariant under rotations a kochen specker theorem for the unsharp spin observables follows there are finite sets of directions such that not all the unsharp spin observables in these directions can consistently be assigned approximate truth values in a noncontextual way

non bicolourable finite configurations of rays and their deformations a new infinite family of examples of finite non bicolourable configurations of rays in hilbert space is described such configurations appear in the analysis of quantum mechanics in terms of bell s inequalities and the kochen specker theorem and illustrate that there is no measurable space in the background of the probability model of a quantum system the mentioned examples are naturally parametrized by a positive integer divisible by 4 and by several complex valued parameters whose number depends on this integer in order to compare two configurations with the same number of rays a notion of deformation of a configuration is introduced the constructed examples are then interpreted as obtained by way of deformations

linear response susceptibility and resonances in chaotic toy models abstract we consider simple examples illustrating some new features of the linear response theory developed by ruelle for dissipative and chaotic systems d ruelle smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics j stat phys 95 1999 393 468 in this theory the concepts of linear response susceptibility and resonance which are familiar to physicists have been revisited due to the dynamical contraction of the whole phase space onto attractors in particular the standard framework of the fluctuation dissipation theorem breaks down and new resonances can show up outside the power spectrum in previous papers we proposed and used new numerical methods to demonstrate the presence of the new resonances predicted by ruelle in a model of chaotic neural networks in this article we deal with simpler models which can be worked out analytically in order to gain more insights into the genesis of the stable resonances and their consequences on the linear response of the system we consider a class of two dimensional time discrete maps describing simple rotator models with a contracting radial dynamics onto the unit circle and a chaotic angular dynamics t 1 2 t mod 2 a generalisation of this system to a network of interconnected rotators is also analysed and related with our previous studies b cessac j a sepulchre stable resonances and signal propagation in a chaotic network of coupled units phys rev e 70 2004 056111 b cessac j a sepulchre transmitting a signal by amplitude modulation in a chaotic network chaos 16 2006 013104 1 13104 12 these models permit us to classify the different types of resonances in the susceptibility and to discuss in particular the relation between the relaxation time of the system to equilibrium with the mixing time given by the decay of the correlation functions also it enables one to propose some general mechanisms responsible for the creation of stable resonances with arbitrary frequencies widths and dependence on the pair of perturbed observed variables

noncontextual hidden variables and physical measurements meyer phys rev lett 83 3751 1999 recently showed that the kochen specker theorem which demonstrates the impossibility of a deterministic hidden variable description of ideal spin measurements on a spin 1 particle can be effectively nullified if only finite precision measurements are considered we generalize this result one can ascribe consistent outcomes to a dense subset of the set of projection valued measurements or to a dense subset of the set of positive operator valued measurements on any finite dimensional system hence no kochen specker like contradiction can rule out hidden variable theories indistinguishable from quantum theory by finite precision measurements in either class

global conservative solutions of the camassa holm equation this paper develops a new approach in the analysis of the camassa holm equation by introducing a new set of independent and dependent variables the equation is transformed into a semilinear system whose solutions are obtained as fixed points of a contractive transformation these new variables resolve all singularities due to possible wave breaking returning to the original variables we obtain a semigroup of global solutions depending continuously on the initial data our solutions are conservative in the sense that the total energy equals a constant for almost every time

differentiation of srb states abstract let f be a diffeomorphism of a manifold m and f a generalized srb state for f if supp f is a hyperbolic compact set we show that the map is differentiable in a suitable functional setup and we compute the derivative when supp f is an attractor the derivative is given by where x is the vector field this formula can be extended to time dependent situations and also at least formally to nonuniformly hyperbolic situations the above results will find their use in the study of the onsager reciprocity relations and the fluctuation dissipation formula of nonequilibrium statistical mechanics

well posedness and peakons for a higher order μ camassa holm equation abstract in this paper we study the cauchy problem of a higher order camassa holm equation by employing the green s function of x 2 2 we obtain the explicit formula of the inverse function x 2 2 w and local well posedness for the equation in sobolev spaces h s s s 7 2 then we prove the existence of global strong solutions and weak solutions moreover we show that the data to solution map is holder continuous in h s s s 4 equipped with the h r s topology for 0 r s finally the equation is shown to admit single peakon solutions which have continuous second derivatives and jump discontinuities in the third derivatives

resonances in a chaotic attractor crisis of the lorenz flow local bifurcations of stationary points and limit cycles have successfully been characterized in terms of the critical exponents of these solutions lyapunov exponents and their associated covariant lyapunov vectors have been proposed as tools for supporting the understanding of critical transitions in chaotic dynamical systems however it is in general not clear how the statistical properties of dynamical systems change across a boundary crisis during which a chaotic attractor collides with a saddle this behavior is investigated here for a boundary crisis in the lorenz flow for which neither the lyapunov exponents nor the covariant lyapunov vectors provide a criterion for the crisis instead the convergence of the time evolution of probability densities to the invariant measure governed by the semigroup of transfer operators is expected to slow down at the approach of the crisis such convergence is described by the eigenvalues of the generator of this semigroup which can be divided into two families referred to as the stable and unstable ruelle pollicott resonances respectively the former describes the convergence of densities to the attractor or escape from a repeller and is estimated from many short time series sampling the state space the latter is responsible for the decay of correlations or mixing and can be estimated from a long times series invoking ergodicity it is found numerically for the lorenz flow that the stable resonances do approach the imaginary axis during the crisis as is indicative of the loss of global stability of the attractor on the other hand the unstable resonances and a fortiori the decay of correlations do not flag the proximity of the crisis thus questioning the usual design of early warning indicators of boundary crises of chaotic attractors and the applicability of response theory close to such crises

a mathematical analysis of the effects of hebbian learning rules on the dynamics and structure of discrete time random recurrent neural networks we present a mathematical analysis of the effects of hebbian learning in random recurrent neural networks with a generic hebbian learning rule including passive forgetting and different timescales for neuronal activity and learning dynamics previous numerical work has reported that hebbian learning drives the system from chaos to a steady state through a sequence of bifurcations here we interpret these results mathematically and show that these effects involving a complex coupling between neuronal dynamics and synaptic graph structure can be analyzed using jacobian matrices which introduce both a structural and a dynamical point of view on neural network evolution furthermore we show that sensitivity to a learned pattern is maximal when the largest lyapunov exponent is close to 0 we discuss how neural networks may take advantage of this regime of high functional interest

approximation of stationary statistical properties of dissipative dynamical systems time discretization we consider temporal approximation of stationary statistical properties of dissipative infinite dimensional dynamical systems we demonstrate that stationary statistical properties of the time discrete approximations i e numerical scheme converge to those of the underlying continuous dissipative infinite dimensional dynamical system under three very natural assumptions as the time step approaches zero the three conditions that are sufficient for the convergence of the stationary statistical properties are 1 uniform dissipativity of the scheme in the sense that the union of the global attractors for the numerical approximations is pre compact in the phase space 2 convergence of the solutions of the numerical scheme to the solution of the continuous system on the unit time interval 0 1 uniformly with respect to initial data from the union of the global attractors 3 uniform continuity of the solutions to the continuous dynamical system on the unit time interval 0 1 uniformly for initial data from the union of the global attractors the convergence of the global attractors is established under weaker assumptions an application to the infinite prandtl number model for convection is discussed

finite element approach to clustering of multidimensional time series we present a new approach to clustering of time series based on a minimization of the averaged clustering functional the proposed functional describes the mean distance between observation data and its representation in terms of mathbf k abstract models of a certain predefined class not necessarily given by some probability distribution for a fixed time series x t this functional depends on mathbf k sets of model parameters theta theta 1 dots theta mathbf k and mathbf k functions of cluster affiliations gamma gamma 1 t dots gamma mathbf k t characterizing the affiliation of any element x t of the analyzed time series to one of the mathbf k clusters defined by the considered model parameters we demonstrate that for a fixed set of model parameters theta the appropriate tykhonov type regularization of this functional with some regularization factor epsilon 2 results in a minimization problem similar to a variational problem usually associated with one dimensional nonhomogeneous partial differential equations this analogy allows us to apply the finite element framework to the problem of time series analysis and to propose a numerical scheme for time series clustering we investigate the conditions under which the proposed scheme allows a monotone improvement of the initial parameter guess with respect to the minimization of the discretized version of the regularized functional we also discuss the interpretation of the regularization factor in the markovian case and show its connection to metastability and exit times the computational performance of the resulting method is investigated numerically on multidimensional test data and is applied to the analysis of multidimensional historical stock market data

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

test models for improving filtering with model errors through stochastic parameter estimation the filtering skill for turbulent signals from nature is often limited by model errors created by utilizing an imperfect model for filtering updating the parameters in the imperfect model through stochastic parameter estimation is one way to increase filtering skill and model performance here a suite of stringent test models for filtering with stochastic parameter estimation is developed based on the stochastic parameterization extended kalman filter spekf these new spekf algorithms systematically correct both multiplicative and additive biases and involve exact formulas for propagating the mean and covariance including the parameters in the test model a comprehensive study is presented of robust parameter regimes for increasing filtering skill through stochastic parameter estimation for turbulent signals as the observation time and observation noise are varied and even when the forcing is incorrectly specified the results here provide useful guidelines for filtering turbulent signals in more complex systems with significant model errors

on the weak solutions to a shallow water equation we obtain the existence of global in time weak solutions to the cauchy problem for a one dimensional shallow water equation that is formally integrable and can be obtained by approximating directly the hamiltonian for euler s equation in the shallow water regime the solution is obtained as a limit of viscous approximation the key elements in our analysis are some new a priori one sided supernorm and space time higher norm estimates on the first order derivatives 2000 john wiley sons inc

a convergent finite difference scheme for the camassa holm equation with general h 1 initial data we suggest a finite dfference scheme for the camassa holm equation that can handle general h 1 initial data the form of the difference scheme is judiciously chosen to ensure that it satisfies a total energy inequality we prove that the difference scheme converges strongly in h 1 towards a dissipative weak solution of the camassa holm equation

global conservative solutions of the generalized hyperelastic rod wave equation abstract we prove existence of global and conservative solutions of the cauchy problem for the nonlinear partial differential equation u t u x x t f u x f u x x x g u 1 2 f u u x 2 x 0 where f is strictly convex or concave and g is locally uniformly lipschitz this includes the camassa holm equation f u u 2 2 and g u u u 2 as well as the hyperelastic rod wave equation f u u 2 2 and g u 3 u 2 2 as special cases it is shown that the problem is well posed for initial data in h 1 r if one includes a radon measure that corresponds to the energy of the system with the initial data the solution is energy preserving stability is proved both with respect to initial data and the functions f and g the proof uses an equivalent reformulation of the equation in terms of lagrangian coordinates

differentiating the absolutely continuous invariant measure of an interval map f with respect to f let the map f 1 1 1 1 have a c i m absolutely continuous f invariant measure with respect to lebesgue let be the change of corresponding to a perturbation x f f 1 of f formally we have for differentiable a

kochen specker vectors some reference citations in the text of this paper are incorrect and should be amended as listed below the reference list is correct as published page 1579 line 5 23 34 38 39 should read 23 34 35 36 page 1579 line 9 40 should read 37 page 1580 line 8 31 33 35 should read 31 33 38 page 1580 line 14 36 should read 39 page 1580 line 36 36 should read 39 page 1580 line 38 37 should read 40 page 1580 line 42 38 should read 35 page 1581 line 3 of caption 33 35 should read 33 38 page 1583 line 12 41 should read 42 page 1587 line 9 39 should read 36 page 1587 line 11 39 should read 36 page 1587 line 18 38 39 should read 35 36 page 1587 line 37 39 should read 36 page 1589 line 18 39 46 47 should read 36 46 47 page 1590 line 2 39 should read 36 age 1590 line 3 39 should read 36

parity proofs of the kochen specker theorem based on the lie algebra e8 the 240 root vectors of the lie algebra e8 lead to a system of 120 rays in a real eight dimensional hilbert space that contains a large number of parity proofs of the kochen specker ks theorem after introducing the rays in a triacontagonal representation due to coxeter we present their ks diagram in the form of a basis table showing all 2025 bases i e sets of eight mutually orthogonal rays formed by the rays only a few of the bases are actually listed but simple rules are given based on the symmetries of e8 for obtaining all the other bases from the ones shown the basis table is an object of great interest because all the parity proofs of e8 can be exhibited as subsets of it we show how the triacontagonal representation of e8 facilitates the identification of substructures that are more easily searched for their parity proofs we have found hundreds of different types of parity proofs ranging from 9 bases or contexts at the low end to 35 bases at the high end and involving projectors of various ranks and multiplicities after giving an overview of the proofs we found we present a few concrete examples of the proofs that illustrate both their generic features as well as some of their more unusual properties in particular we present a proof involving 34 rays and 9 bases that appears to provide the most compact proof of the ks theorem found to date in eight dimensions

parametric smoothness and self scaling of the statistical properties of a minimal climate model what beyond the mean field theories abstract a quasi geostrophic intermediate complexity model of the mid latitude atmospheric circulation is considered featuring simplified baroclinic conversion and barotropic convergence processes the model undergoes baroclinic forcing towards a given latitudinal temperature profile controlled by the forced equator to pole temperature difference t e as t e increases a transition takes place from a stationary regime hadley equilibrium to a periodic regime and eventually to a chaotic regime where evolution takes place on a strange attractor the attractor dimension metric entropy and bounding box volume in phase space have a smooth dependence on t e which results in power law scaling properties power law scalings with respect to t e are detected also for the statistical properties of global physical observables the total energy of the system and the averaged zonal wind the scaling laws which constitute the main novel result of the present work can be thought to result from the presence of a statistical process of baroclinic adjustment which tends to decrease the equator to pole temperature difference and determines the properties of the attractor of the system the self similarity could be of great help in setting up a theory for the overall statistical properties of the general circulation of the atmosphere and in guiding on a heuristic basis both data analysis and realistic simulations going beyond the unsatisfactory mean field theories and brute force approaches a leading example for this would be the possibility of estimating the sensitivity of the output of the system with respect to changes in the parameters

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

heterogeneous multiscale methods for stiff ordinary differential equations the heterogeneous multiscale methods hmm is a general framework for the numerical approximation of multiscale problems it is here developed for ordinary differential equations containing differe

filtering a statistically exactly solvable test model for turbulent tracers from partial observations a statistically exactly solvable model for passive tracers is introduced as a test model for the authors nonlinear extended kalman filter nekf as well as other filtering algorithms the model involves a gaussian velocity field and a passive tracer governed by the advection diffusion equation with an imposed mean gradient the model has direct relevance to engineering problems such as the spread of pollutants in the air or contaminants in the water as well as climate change problems concerning the transport of greenhouse gases such as carbon dioxide with strongly intermittent probability distributions consistent with the actual observations of the atmosphere one of the attractive properties of the model is the existence of the exact statistical solution in particular this unique feature of the model provides an opportunity to design and test fast and efficient algorithms for real time data assimilation based on rigorous mathematical theory for a turbulence model problem with many active spatiotemporal scales here we extensively study the performance of the nekf which uses the exact first and second order nonlinear statistics without any approximations due to linearization the role of partial and sparse observations the frequency of observations and the observation noise strength in recovering the true signal its spectrum and fat tail probability distribution are the central issues discussed here the results of our study provide useful guidelines for filtering realistic turbulent systems with passive tracers through partial observations