stretching based diagnostics and reduction of chemical kinetic models with diffusion a new method for diagnostics and reduction of dynamical systems and chemical kinetic models is proposed the method makes use of the local structure of the normal stretching rates by projecting the dynamics onto the local directions of maximal stretching the approach is computationally very simple as it implies the spectral analysis of a symmetric matrix notwithstanding its simplicity stretching based analysis derives from a geometric basis grounded on the pointwise applications of concepts of normal hyperbolicity theory as a byproduct a simple reduction method is derived equivalent to a local embedding algorithm which is based on the local projection of the dynamics onto the most unstable and or slow modes compared to the time scale dictated by the local tangential dynamics this method provides excellent results in the analysis and reduction of dynamical systems displaying relaxation towards an equilibrium point limit cycles and chaotic attractors several numerical examples deriving from typical models of reaction diffusion kinetics exhibiting complex dynamics are thoroughly addressed the application to typical combustion models is also analyzed

model reduction via parametrized locally invariant manifolds some examples a method for model reduction in non linear ode systems is demonstrated through computational examples the method does not require an implicit separation of time scales in the fine dynamics to be effective from the computational standpoint the method has the potential of serving as a subgrid modeling tool from the physical standpoint it provides a model for interpreting and describing history dependence in coarse grained response of an autonomous system

new applications of numerical simulation based on lattice boltzmann method at high reynolds numbers abstract in order to study the flow behavior at high reynolds numbers two modified models known as the multiple relaxation time lattice boltzmann method mrt lbm and large eddy simulation lattice boltzmann method les lbm have been employed in this paper the mrt lbm was designed to improve numerical stability at high reynolds numbers by introducing multiple relaxation time terms which consider the variations of density energy momentum energy flux and viscous stress tensor as a result mrt lbm is capable of dealing with turbulent flows considering energy dispersion and dissipation in the present paper this model was employed to simulate the flow at turbulent reynolds numbers in wall driven cavities two sided wall driven cavity flow was studied for the first time based on mrt lbm at reynolds numbers ranging from 2 1 0 4 to 1 1 0 6 and employing a very large resolution2048 2048 it is found that whenever top and bottom lids are moving in the opposite directions and the reynolds number is higher than 2 1 0 4 the flow is chaotic although some quasi symmetric properties still remain fully disappearing at reynolds numbers between 2 1 0 5 and 3 1 0 5 furthermore between this reynolds numbers range 2 1 0 5 re 3 1 0 5 the quasi symmetric structures turn into a much smaller and fully chaotic eddies the les lbm model implements the large eddy simulation turbulent model into the conventional lbm allowing to study the flow at turbulent reynolds numbers les lbm combined with quadruple tree cartesian cutting grid tree grid was employed for the first time to characterize the flow dynamics over a cylinder and a hump at relatively high reynolds numbers in order to construct the macroscopic quantities in the virtual boundaries separating two different grid levels a set of new schemes were designed the coupling of the les lbm and tree grid drastically reduced the computational time required to perform the simulations thus allowing to minimize the hardware requirements les lbm model is shown to be much more efficient when combined with the tree grid instead of using the standard cartesian grid

entropic lattice boltzmann method for microflows abstract a new method for the computation of flows at the micrometer scale is presented it is based on the recently introduced minimal entropic kinetic models both the thermal and isothermal families of minimal models are presented and the simplest isothermal entropic lattice bhatnagar gross krook elbgk is studied in detail in order to quantify its relevance for microflow simulations elbgk is equipped with boundary conditions which are derived from molecular models diffusive wall a map of three dimensional kinetic equations onto two dimensional models is established which enables two dimensional simulations of quasi two dimensional flows the elbgk model is studied extensively in the simulation of the two dimensional poiseuille channel flow results are compared to known analytical and numerical studies of this flow in the setting of the bhatnagar gross krook model the elbgk is in quantitative agreement with analytical results in the domain of weak rarefaction characterized by knudsen number kn the ratio of mean free path to the hydrodynamic scale up to kn 0 01 which is the domain of many practical microflows moreover the results qualitatively agree throughout the entire knudsen number range demonstrating knudsen s minimum for the mass flow rate at moderate values of kn as well as the logarithmic scaling at large kn the present results indicate that elbm can complement or even replace computationally expensive microscopic simulation techniques such as kinetic monte carlo and or molecular dynamics for low mach and low knudsen number hydrodynamics pertinent to microflows

remarks on the global sobolev inequalities in the minkowski space rn 1 on demontre des inegalites de sobolev globales sur l espace de minkowski r n 1 le theoreme principal est valable sur des tranches a temps fixe

quasi equilibrium grid algorithm geometric construction for model reduction the method of invariant grid mig is an iterative procedure for model reduction in chemical kinetics which is based on the notion of slow invariant manifold sim a n gorban i v karlin method of invariant manifold for chemical kinetics chem eng sci 58 2003 4751 4768 e chiavazzo a n gorban i v karlin comparison of invariant manifolds for model reduction in chemical kinetics commun comput phys 2 5 2007 964 992 a n gorban i v karlin a y zinovyev invariant grids for reaction kinetics physica a 333 2004 106 154 a n gorban i v karlin invariant manifolds for physical and chemical kinetics lecture notes physics 660 springer berlin heidelberg 2005 doi 10 1007 b98103 important role in that method is played by the initial grid which once refined gives a description of the invariant manifold the invariant grid a convenient way to get a first approximation of the sim is given by the spectral quasi equilibrium manifold sqem a n gorban i v karlin method of invariant manifold for chemical kinetics chem eng sci 58 2003 4751 4768 e chiavazzo a n gorban i v karlin comparison of invariant manifolds for model reduction in chemical kinetics commun comput phys 2 5 2007 964 992 in the present paper a flexible numerical method to construct the discrete analog of a quasi equilibrium manifold in any dimension is presented that object is named quasi equilibrium grid qeg while the procedure quasi equilibrium grid algorithm qega extensions of the qem notion are also suggested the qega is a numerical tool which can be used to find a grid based approximation for the locus of minima of a convex function under some linear constraints the method is validated by construction of one and two dimensional grids for a model of hydrogen oxidation reaction

density matrix renormalization for model reduction in nonlinear dynamics we present an approach for model reduction of nonlinear dynamical systems based on proper orthogonal decomposition pod our method derived from the density matrix renormalization group provides a significant reduction in computational effort for the calculation of the reduced system compared to a pod the efficiency of the algorithm is tested on the one dimensional burgers equations and a one dimensional equation of the fisher type as nonlinear model systems

invariant grids for reaction kinetics abstract in this paper we construct low dimensional manifolds of reduced description for equations of chemical kinetics from the standpoint of the method of invariant manifold mim mim is based on a formulation of the condition of invariance as an equation and its solution by newton iterations a grid based version of mim is developed the method of invariant grids we describe the newton method and the relaxation method for the invariant grids construction the problem of the grid correction is fully decomposed into the problems of the grid s nodes correction the edges between the nodes appear only in the calculation of the tangent spaces this fact determines high computational efficiency of the method of invariant grids the method is illustrated by two examples the simplest catalytic reaction michaelis menten mechanism and the hydrogen oxidation the algorithm of analytical continuation of the approximate invariant manifold from the discrete grid is proposed generalizations to open systems are suggested the set of methods covered makes it possible to effectively reduce description in chemical kinetics

numerical taylor expansions for invariant manifolds we consider numerical computation of taylor expansions of invariant manifolds around equilibria of maps and flows these expansions are obtained by writing the corresponding functional equation in a number of points setting up a nonlinear system of equations and solving this system using a simplified newton s method this approach will avoid symbolic or explicit numerical differentiation the linear algebra issues of solving the resulting sylvester equations are studied in detail

global existence of null form wave equations on small asymptotically euclidean manifolds abstract we prove the global existence of the small solutions to the cauchy problem for quasilinear wave equations satisfying the null condition on r 3 g where the metric g is a small perturbation of the flat metric and approaches the euclidean metric like 1 x 2 2 with 1 global and almost global existence for systems without the null condition are also discussed for certain small time dependent perturbations of the flat metric in appendix a

the effect of lumping and expanding on kinetic differential equations let us consider the differential equation dot bf y t bf f bf y t with an bf f from bf r n to bf r n and suppose that there exists a transformation bf h from bf r n to bf r hat n hat n le n such that hat bf y bf h circ bf y obeys a differential equation dot hat bf y t hat bf f hat bf y t with some function hat bf f then the first equation is said to be lumpable to the second by bf h here mainly the case is investigated when the original differential equation has been induced by a complex chemical reaction we provided a series of necessary and sufficient conditions for the existence of such functions bf h and hat bf f some of them are formulated in terms of bf h and bf f only beyond these conditions our main concern here is how lumping changes those properties of the solutions which are either interesting from the point of view of the qualitative theory of differential equations or from the point

dynamic correction to moment approximations another question comes from the approximate character of the grad equations and is discussed in frames of the eit while the grad equations are strictly hyperbolic at any level n i e predicting a finite speed of propagation will this feature will be preserved in the further corrections these two questions are special cases of a more general one namely how does one derive a closed description with a given number of moments such a description is sometimes called mesoscopic 7 since it occupies an intermediate level between the hydrodynamic macroscopic and the kinetic microscopic levels of description in this paper we aim at deriving the mesoscopic dynamics of thirteen moments in the simplest case when the kinetic description satisfies the linearized boltzmann equation our approach will be based on the two assumptions i the mesoscopic dynamics of thirteen moments exists and is invariant with respect to the microscopic dynamics ii the 13m grad approximation is a suitable first approximation to this mesoscopic dynamics the assumption i is realized as the invariance equation for the unknown mesoscopic distribution function following the assumption ii we solve the invariance equation iteratively taking the 13m grad approximation for the input approximation and consider the first iteration further we refer to this as to the dynamic correction to distinguish from constructing another ansatz w e demonstrate that the correction results in the exact ce transport coefficients we also demonstrate how the dynamic correction modifies the hyperbolicity of the grad equations a similar viewpoint on derivation of hydrodynamics was earlier developed in 8 we will return to a comparison below first we review the boltzmann equation and the 13m grad approximation we denote as n0 u050 and p0 the equilibrium values of the hydrodynamic parameters n is the number density u is the average velocity and p5nkbt is the pressure the global maxwell distribution function f is

relaxational trajectories global approximations abstract the paper intends to fill the gap of analytic approximate methods for non linear space independent dissipative systems equipped with the entropy functional the key point of the analysis is an upper limiting state in the beginning of the relaxation extremal properties of this state are described and explicit estimations are derived this limiting state is used to construct explicit approximations of the phase trajectories special attention is paid to accomplish positivity smoothness and the entropy growth along the approximate trajectories the method is tested for the space independent boltzmann equation with various collisional mechanisms

maximum of the local entropy production becomes minimal in stationary processes in this paper we propose a new principle for stationary thermodynamic processes the maximum of the local entropy production becomes minimal in stationary processes in order to show the usefulness of the principle we consider one dimensional stationary heat transfer in monatomic gases here we solve extended moment schemes that follow from the boltzmann equation such schemes require boundary conditions for all moments under consideration and these are constructed by means of the new principle moreover we show that the minimum principle for the global entropy production does not lead to good results in this case s0031 9007 98 06306 6

lattice boltzmann method with selective viscosity filter for high reynolds number flows the lattice boltzmann method suffers from numerical instabilities that can induce local blowup of the computation the von neumann stability analysis applied to the lbe bgk and lbe mrt models shows that numerical instabilities occur in the high wavenumber range and are due to the interplay between acoustic modes and some other modes as it is done in the lbe mrt model an increase of the bulk viscosity is an efficient way of damping spurious oscillations however this stabilization method induces an over damping of acoustic waves some selective spatial filters can be used in order to eliminate the spurious small spatial scales without affecting the large scale physical modes three different lattice boltzmann algorithms based on filtering step are proposed the fully filtered lbe the lbe with filtered macroscopic quantities and the lbe with filtered collision operator the behavior of several explicit filter stencils is studied in the fourier space for a given filter stencil the filtered collision operator approach leads to the highest cut off wavenumber in this case the theoretical wavenumber dependent viscosity is n k c s 2 t email protected k 1 2 where f k is the filter shape and s the filter strength under resolved simulations high reynolds number are performed on the case of the doubly periodic shear layers the performance of the three filtered lbe is found to be the same as the mrt model for stability control propagation of acoustic plane waves is also simulated with the three filtering algorithms the measured dissipation of acoustic wave compares well with the theoretical results

fast and slow invariant manifolds in chemical kinetics abstract in the framework of the classical theory of singularly perturbed systems sps there exist two main types of invariant manifolds i e fast and slow manifolds with their own intrinsic dynamics there are many different methods for dimension and complexity reduction for mechanisms of chemical kinetics based on the concept of slow invariant manifolds for instance most known and popular techniques such as quasi steady state qssa and partial equilibrium approaches pea as well as recently developed intrinsic low dimensional manifolds ildm and computational singular perturbation csp methods have shown a potential to treat automatically very complex mechanisms comprising hundreds of chemically reacting species in this respect the importance of slow invariant manifolds has become more or less clear however the importance of fast invariant manifolds is much less transparent though without an accurate knowledge about fast manifolds fast fibers in other terminology slow manifolds cannot be appropriately identified in this paper we shall discuss a concept of singularly perturbed vector fields spvf that is a coordinate free theory of singularly perturbed systems with the main emphasis made on fast invariant manifolds and on the method which allows identification and evaluation of these objects the importance of these objects in model reduction of mechanisms of chemical kinetics will be addressed motivation and application of the method will be demonstrated by a simple model example and by analysis of conventional qssa and pea methods

the quasi steady state assumption a case study in perturbation the quasi steady state assumption qssa of biochemistry is studied as an approximation that is important in itself and also that exemplifies an approach to ode systems with an initial fast transient simple estimates of the two relevant time scales of the underlying problem are made these estimates lead to a dimensionless formulation containing a different small parameter from that which is customary earlier results on the qssa are surveyed in the context of the new nondimensionalization it is shown that more comprehensive and simpler error estimates can now be made some general methodological hints are drawn from this example

the formula isomorphism problem we investigate the computational complexity of the formula isomorphism problem fi on input of two boolean formulas f and g decide whether there exists a permutation of the variables of g such that f and g become equivalent fi is contained in sigma 2 bf p the second level of the polynomial hierarchy our main result is a one round interactive proof for the complementary formula nonisomorphism problem fni where the verifier has access to an np oracle to obtain this we use a result from learning theory by bshouty et al that boolean formulas can be learned probabilistically with equivalence queries and access to an np oracle as a consequence fi cannot be sigma 2 bf p complete unless the polynomial hierarchy collapses further properties of fi are shown fi has and and or functions the counting version fi can be computed in polynomial time relative to fi and fi is self reducible

c1approximations of inertial manifolds for dissipative nonlinear equations abstract in this paper we study a class of nonlinear dissipative partial differential equations that have inertial manifolds this means that the long time behavior is equivalent to a certain finite system of ordinary differential equations we investigate ways in which these finite systems can be approximated in thec1sense geometrically this may be interpreted as constructing manifolds in phase space that arec1close to the inertial manifold of the partial differential equation under such approximations the invariant hyperbolic sets of the global attractor persist

approximation of slow and fast dynamics in multiscale dynamical systems by the linearized relaxation redistribution method in this paper we introduce a fictitious dynamics for describing the only fast relaxation of a stiff ordinary differential equation ode system towards a stable low dimensional invariant manifold in the phase space slow invariant manifold sim as a result the demanding problem of constructing sim of any dimensions is recast into the remarkably simpler task of solving a properly devised ode system by stiff numerical schemes available in the literature in the same spirit a set of equations is elaborated for local construction of the fast subspace and possible initialization procedures for the above equations are discussed the implementation to a detailed mechanism for combustion of hydrogen and air has been carried out while a model with the exact chapman enskog solution of the invariance equation is utilized as a benchmark

exact summation of the chapman enskog expansion from moment equations the nonlinear recurrence chapman enskog solution for the linearized grad ten moment equations is resummed exactly using this solution the stability of higher order hydrodynamics in various approximations is discussed

almost global existence to nonlinear wave equations in three space dimensions on considere l existence a t grand de solutions de petite amplitude pour des equations d onde d ordre 2 non lineaires a 4 dimensions d espace temps

general approach to constructing models of the boltzmann equation abstract the problem of thermodynamic parameterization of an arbitrary approximation of reduced description is solved on the base of this solution a new class of model kinetic equations is constructed that gives a model extension of the chosen approximation to a kinetic model model equations describe two processes rapid relaxation to the chosen approximation along the planes of rapid motions and the slow motion caused by the chosen approximation the h theorem is proved for these models it is shown that the rapid process always leads to entropy growth and also a neighborhood of the approximation is determined inside which the slow process satisfies the h theorem kinetic models for grad moment approximations and for the tamm mott smith approximation are constructed explicitly in particular the problem of concordance of the es model with the h theorem is solved

the lifespan of a class of smooth spherically symmetric solutions of the compressible euler equations with variable entropy in three space dimensions we consider smooth three dimensional spherically symmetric eulerian flows of ideal polytropic gases with variable entropy whose initial data are obtained by adding a small smooth perturbation with compact support to a constant state under a natural assumption we obtain precise information on the asymptotic behavior of their lifespan when the size of the initial perturbation tends to 0 this is achieved by the construction and estimate of a suitable approximate flow

entropy function approach to the lattice boltzmann method in this paper we present the construction of the lattice boltzmann method equipped with the h theorem based on entropy functions whose local equilibria are suitable to recover the navier stokes equations in the framework of the lattice boltzmann method we derive a collision integral which enables simple identification of transport coefficients and which circumvents construction of the equilibrium we discuss performance of this approach as compared to the standard realizations

taylor approximation of invariant fiber bundles for nonautonomous difference equations abstract invariant fiber bundles generalize invariant manifolds to nonautonomous difference equations in this paper we develop a method to calculate their taylor approximation which is of crucial importance e g for an application of the reduction principle in a nonautonomous setting

existence for large times of strict solutions of nonlinear wave equations in three space dimensions for small initial data on considere le probleme aux valeurs initiales pour certaines equations hyperboliques non lineaires d ordre 2

variational approach to closure of nonlinear dynamical systems autonomous case a general approach for the derivation of nonlinear parameterizations of neglected scales is presented for nonlinear systems subject to an autonomous forcing in that respect dynamically based formulas are derived subject to a free scalar parameter to be determined per mode to parameterize for each high mode this free parameter is obtained by minimizing a cost functional a parameterization defect depending on solutions from direct numerical simulation dns but over short training periods of length comparable to a characteristic recurrence or decorrelation time of the dynamics an important class of dynamically based formulas for our parameterizations to optimize are obtained as parametric variations of manifolds approximating the invariant ones to better appreciate the origins of the modified manifolds thus obtained the standard approximation theory of invariant manifolds is revisited in part i of this article a special emphasis is put on backward forward bf systems naturally associated with the original system whose asymptotic integration provides the leading order approximation of invariant manifolds part ii presents then i the modifications of these approximating manifolds based also on integration of the same bf systems but this time over a finite time tau and ii the variational approach aimed at making an efficient selection of tau per mode to parameterize the parametric class of leading interaction approximation lia of the high modes obtained this way is completed by another parametric class built from the quasi stationary approximation qsa close to the first criticality the qsa is an approximation to the lia but it differs as one moves away from criticality rigorous results are derived that show that given a cutoff dimension the best manifolds that can be obtained through our variational approach are manifolds which are in general no longer invariant the minimizers are objects called the optimal parameterizing manifolds pms that are intimately tied to the conditional expectation of the original system i e the best vector field of the reduced state space resulting from averaging of the unresolved variables with respect to a probability measure conditioned on the resolved variables applications to the closure of low order models of atmospheric primitive equations and rayleigh benard convection are then discussed the approach is finally illustrated in the context of the kuramoto sivashinsky turbulence as providing efficient closures without slaving for a cutoff scale k mathfrak c placed within the inertial range and the reduced state space is just spanned by the unstable modes without inclusion of any stable modes whatsoever the underlying optimal pms obtained by our variational approach are far from slaving and allow for remedying the excessive backscatter transfer of energy to the low modes encountered by the lia or the qsa parameterizations in their standard forms when they are used at this cutoff wavelength

inference of analytical thermodynamic models for biological networks abstract we present an automated algorithm for inferring analytical models of closed reactive biochemical mixtures on the basis of standard approaches borrowed from thermodynamics and kinetic theory of gases as input the method requires a number of steady states i e an equilibria cloud in phase space and at least one time series of measurements for each species validations are discussed for both the michaelis menten mechanism four species two conservation laws and the mitogen activated protein kinase mapk mechanism eleven species three conservation laws

local dissipative hamiltonian description of reversible reaction networks in this letter we show that closed reversible chemical reaction networks with independent elementary reactions admit a global pseudo hamiltonian structure which is at least locally dissipative around any equilibrium point the structure matrix of the hamiltonian description reflects the graph topology of the reaction network and it is a smooth function of the concentrations of the chemical species in the positive orthant the physical interpretation of the description is briefly explained and two illustrative examples are presented for global and local dissipative hamiltonian description respectively

uniqueness of thermodynamic projector and kinetic basis of molecular individualism three results are presented first we solve the problem of persistence of dissipation for reduction of kinetic models kinetic equations with thermodynamic lyapunov functions are studied uniqueness of the thermodynamic projector is proven there exists only one projector which transforms any vector field equipped with the given lyapunov function into a vector field with the same lyapunov function for a given anzatz manifold which is not tangent to the lyapunov function levels second we use the thermodynamic projector for developing the short memory approximation and coarse graining for general nonlinear dynamic systems we prove that in this approximation the entropy production increases the theorem about entropy overproduction in example we apply the thermodynamic projector to derive the equations of reduced kinetics for the fokker planck equation a new class of closures is developed the kinetic multipeak polyhedra distributions of this type are expected in kinetic models with multidimensional instability as universally as the gaussian distribution appears for stable systems the number of possible relatively stable states of a nonequilibrium system grows as 2m and the number of macroscopic parameters is in order mn where n is the dimension of configuration space and m is the number of independent unstable directions in this space the elaborated class of closures and equations pretends to describe the effects of molecular individualism this is the third result

numerical taylor expansions of invariant manifolds in large dynamical systems in this paper we develop a numerical method for computing higher order local approximations of invariant manifolds such as stable unstable or center manifolds near steady states of a dynamical system the underlying system is assumed to be large in the sense that a large sparse jacobian at the equilibrium occurs for which only a linear black box solver and a low dimensional invariant subspace is available but for which methods like the qr algorithm are considered to be too expensive our method is based on an analysis of the multilinear sylvester equations for the higher derivatives which can be solved under certain nonresonance conditions these conditions are weaker than the standard gap conditions on the spectrum which guarantee the existence of the invariant manifold the final algorithm requires the solution of several large linear systems with a bordered jacobian to these systems we apply a block elimination method recently developed by govaerts and pryce 12 14

continuation techniques and interactive software for bifurcation analysis of odes and iterated maps physics abstract we present a numerical technique for the analysis of local bifurcations which is based on the continuation of structurally unstable invariant sets in a suitable phase parameter space the invariant sets involved in our study are equilibrium points and limit cycles of autonomous odes periodic solutions of time periodic nonautonomous odes fixed points and periodic orbits of iterated maps the more general concept of a continuation strategy is also discussed it allows the analysis of various singularities of generic systems and of their mutual relationships the approach is extended to codimension three singularities we introduce several bifurcation functions and show how to use them to construct well posed continuation problems the described continuation technique is supported by an interactive graphical program called locbif we discuss briefly the concepts of the locbif interface and give some examples of typical applications

model complexity reduction of chemical reaction networks using mixed integer quadratic programming the model complexity reduction problem of large chemical reaction networks under isobaric and isothermal conditions is considered with a given detailed kinetic mechanism and measured data of the key species over a finite time horizon the complexity reduction is formulated in the form of a mixed integer quadratic optimization problem where the objective function is derived from the parametric sensitivity matrix the proposed method sequentially eliminates reactions from the mechanism and simultaneously tunes the remaining parameters until the pre specified tolerance limit in the species concentration space is reached the computational efficiency and numerical stability of the optimization are improved by a pre reduction step followed by suitable scaling and initial conditioning of the hessian involved the proposed complexity reduction method is illustrated using three well known case studies taken from reaction kinetics literature 2012 elsevier ltd all rights reserved

macroscopic constraints for the minimum entropy production principle in an essential and quite general setup based on networks we identify schnakenberg s observables as the constraints that prevent a system from relaxing to equilibrium showing that in the linear regime steady states satisfy a minimum entropy production principle the result is applied to master equation systems opening a new path to a well known version of the principle regarding invariant states moreover with the aid of a simple example the principle is shown to conform to prigogine s original formulation finally we discuss analogies and differences with a recently proposed maximum entropy production principle

constant depth circuits fourier transform and learnability in this paper boolean functions in 4c0 are studied using harmonic analysis on the cube the main result is that an aco boolean function has almost all of its power spectrum on the low order coefficients an important ingredient of the proof is hastad s switching lemma 8 this result implies several new properties of functions in 4c functions in ac have low average sensitivity they may be approximated well by a real polynomial of low degree and they cannot be pseudorandom function generators perhaps the most interesting application is an o n poiyiog n time algorithm for learning func tions in aco the algorithm observes the behavior of an ac function on o npo y og n randomly chosen inputs and derives a good approximation for the fourier transform of the function this approximation allows the algorithm to predict with high probability the value of the function on other randomly chosen inputs

two dimensional lattice boltzmann model for compressible flows with high mach number abstract in this paper we present an improved lattice boltzmann model for compressible navier stokes system with high mach number the model is composed of three components i the discrete velocity model by m watari and m tsutahara phys rev e 67 2003 036306 ii a modified lax wendroff finite difference scheme where reasonable dissipation and dispersion are naturally included iii artificial viscosity the improved model is convenient to compromise the high accuracy and stability the included dispersion term can effectively reduce the numerical oscillation at discontinuity the added artificial viscosity helps the scheme to satisfy the von neumann stability condition shock tubes and shock reflections are used to validate the new scheme in our numerical tests the mach numbers are successfully increased up to 20 or higher the flexibility of the new model makes it suitable for tracking shock waves with high accuracy and for investigating nonlinear nonequilibrium complex systems

the parameterization method for invariant manifolds iii overview and applications abstract we describe a method to establish existence and regularity of invariant manifolds and at the same time to find simple maps which are conjugated to the dynamics on them the method establishes several invariant manifold theorems for instance it reduces the proof of the usual stable manifold theorem near hyperbolic points to an application of the implicit function theorem in banach spaces we also present several other applications of the method

adaptive simplification of complex multiscale systems a fully adaptive methodology is developed for reducing the complexity of large dissipative systems this represents a significant step toward extracting essential physical knowledge from complex systems by addressing the challenging problem of a minimal number of variables needed to exactly capture the system dynamics accurate reduced description is achieved by construction of a hierarchy of slow invariant manifolds with an embarrassingly simple implementation in any dimension the method is validated with the autoignition of the hydrogen air mixture where a reduction to a cascade of slow invariant manifolds is observed

equations d ondes quasilineaires et estimations de strichartz dans cet article notre but est de resoudre des equations d ondes quasilineaires pour des donnees initiales moins regulieres que ce qu imposent les methodes d energie ceci impose de demontrer des estimees de type strichartz pour des operateurs d ondes a coefficients seulement lipschitziens

reduced description in the reaction kinetics models of complex reactions in thermodynamically isolated systems often demonstrate evolution towards low dimensional manifolds in the phase space for this class of models we suggest a direct method to construct such manifolds and thereby to reduce the eective dimension of the problem the approach realizes the invariance principle of the reduced description it is based on iterations rather than on a small parameter expansion it leads to tractable linear problems and is consistent with thermodynamic requirements the approach is tested with a model of catalytic reaction c 2000 elsevier science b v all rights reserved

three dimensional multi relaxation time mrt lattice boltzmann models for multiphase flow in this paper three dimensional 3d multi relaxation time mrt lattice boltzmann lb models for multiphase flow are presented in contrast to the bhatnagar gross krook bgk model a widely employed kinetic model in mrt models the rates of relaxation processes owing to collisions of particle populations may be independently adjusted as a result the mrt models offer a significant improvement in numerical stability of the lb method for simulating fluids with lower viscosities we show through the chapman enskog multiscale analysis that the continuum limit behavior of 3d mrt lb models corresponds to that of the macroscopic dynamical equations for multiphase flow we extend the 3d mrt lb models developed to represent multiphase flow with reduced compressibility effects the multiphase models are evaluated by verifying the laplace young relation for static drops and the frequency of oscillations of drops the results show satisfactory agreement with available data and significant gains in numerical stability

postprocessing the galerkin method a novel approach to approximate inertial manifolds a postprocess of the standard galerkin method for the discretization of dissipative equations is presented the postprocessed galerkin method uses the same approximate inertial manifold phi app to approximate the high wave number modes of the solution as in the nonlinear galerkin nlg method however in this postprocessed galerkin method the value of phi app is calculated only once after the time integration of the standard galerkin method is completed contrary to the nlg in which phi app evolves with time and affects the time evolution of the lower wave number modes the postprocessed galerkin method which is much cheaper to implement computationally than the nlg is shown in the case of fourier modes to possess the same rate of convergence accuracy as the simplest version of the nlg which is based on either the foias manley temam approximate inertial manifold or the euler galerkin approximate inertial manifold this is proved for some problems in one and two spatial dimensions including the navier stokes equations under periodic boundary conditions the advantages of postprocessing that we present here apply not only to the standard galerkin method but also to the computationally more efficient pseudospectral method

continuation and bifurcations numerical techniques and applications bifurcation to rotating waves from non trivial steady states use of approximate inertial manifolds in bifurcation calculations understanding steady state bifurcation diagrams for a model reaction diffusion system bifurcations chaos and self organization in reaction diffusion systems eigenvalue problems with the symmetry of a group and bifurcations steady state steady state mode interaction in nonlinear equations with z2 symmetry symbolic computation and bifurcation methods bifurcation analysis a combined numerical and analytical approach to the numerical solution of symmetry breaking bifurcation problems a computational method and path following for periodic solutions with symmetry global bifurcations and their numerical computation computation of invariant manifold bifurcations a method for homoclinic and heteroclinic continuation in two and three dimensions the global attractor under discretisation the numerical detection of hopf bifurcation points a newton like method for simple bifurcation problems with application to large sparse systems aspects of continuation software interactive system for studies in nonlinear dynamics linlbf a program for continuation and bifurcation analysis of equilibria up to codimension three on the topology of three dimensional separations a guide for classification the construction of cubature formulae using continuation and bifurcation software determining an organizing center for passive optical systems optimization by continuation stability of marangoni convection in a microgravity environment computing with reaction diffusion systems applications in image processing bifurcation of codimension 2 for a discrete map bifurcation of periodic solutions in pde s numerical techniques and applications abstract bifurcation and chaos in chua s circuit family abstract continuation and collocation for parameter dependent boundary value problems abstract porous medium combustion abstract block elimination and the computation of simple turning points abstract bifurcation into gaps in the essential spectrum abstract application of numerical continuation in aerospace problems abstract application of a reduced basis method in structural analysis abstract some applications of bifurcation theory in engineering abstract higher order predictors in numerical path following schemes abstract

hilbert s 6th problem exact and approximate hydrodynamic manifolds for kinetic equations the problem of the derivation of hydrodynamics from the boltz mann equation and related dissipative systems is formulated as the problem of slow invariant manifold in the space of distributions we review a few in stances where such hydrodynamic manifolds were found analytically both as the result of summation of the chapman enskog asymptotic expansion and by the direct solution of the invariance equation these model cases comprising grad s moment systems both linear and nonlinear are studied in depth in order to gain understanding of what can be expected for the boltzmann equa tion particularly the dispersive dominance and saturation of dissipation rate of the exact hydrodynamics in the short wave limit and the viscosity modifica tion at high divergence of the flow velocity are indicated as severe obstacles to the resolution of hilbert s 6th problem furthermore we review the derivation of the approximate hydrodynamic manifold for the boltzmann equation using newton s iteration and avoiding smallness parameters and compare this to the exact solutions additionally we discuss the problem of projection of the boltzmann equation onto the approximate hydrodynamic invariant manifold using entropy concepts finally a set of hypotheses is put forward where we describe open questions and set a horizon for what can be derived exactly or proven about the hydrodynamic manifolds for the boltzmann equation in the future

numerical normalization techniques for all codim 2 bifurcations of equilibria in ode s explicit computational formulas for the coefficients of the normal forms for all codim 2 equilibrium bifurcations of equilibria in autonomous odes are derived they include second order coefficients for the bogdanov takens bifurcation third order coefficients for the cusp and fold hopf bifurcations and coefficients of the fifth order terms for the generalized hopf bautin and double hopf bifurcations the formulas are independent on the dimension of the phase space and involve only critical eigenvectors of the jacobian matrix of the right hand sides and its transpose as well as multilinear functions from the taylor expansion of the right hand sides at the critical equilibrium the normal form coefficients for the fold hopf bifurcation in the new lorenz model are computed using the derived formulas proving the existence of a nontrivial invariant set in the system

dissipative brackets as a tool for kinetic modeling abstract a generalization of the previously introduced dissipative bracket formulation of dissipative kinetic equations is performed this generalization offers an opportunity to write model equations in concordance with the entropy requirements in situations where one attempts to consider the dissipative dynamics near highly non equilibrium states a list of sufficient conditions generalizing both the detailed balance and the stueckelberg unitarity condition is established for the relevant kinetic equations to have an analog of the boltzmann h theorem several specific patterns of an explicit derivation of the model kinetic equations are discussed including the relaxation in a given initial direction the relaxation between two non equilibrium states and the slow decay to the equilibrium

nonlinear observer design using lyapunov s auxiliary theorem abstract the present work proposes a new approach to the nonlinear observer design problem based on the early ideas that influenced the development of the linear luenberger observer theory the proposed approach develops a nonlinear analogue the formulation of the observer design problem is realized via a system of singular first order linear pdes and a rather general set of necessary and sufficient conditions for solvability is derived by using lyapunov s auxiliary theorem the solution to the above system of pdes is locally analytic and this enables the development of a series solution method that is easily programmable with the aid of a symbolic software package

asymptotic analysis of two reduction methods for systems of chemical reactions abstract this paper concerns two methods for reducing large systems of chemical kinetics equations namely the method of intrinsic low dimensional manifolds ildms due to maas and pope combust flame 88 1992 239 and an iterative method due to fraser j chem phys 88 1988 4732 and further developed by roussel and fraser j chem phys 93 1990 1072 both methods exploit the separation of fast and slow reaction time scales to find low dimensional manifolds in the space of species concentrations where the long term dynamics are played out the asymptotic expansions of these manifolds e 0 where e measures the ratio of the reaction time scales are compared with the asymptotic expansion of m e the slow manifold given by geometric singular perturbation theory it is shown that the expansions of the ildm and m e agree up to and including terms of o e the former has an error at o e 2 that is proportional to the local curvature of m 0 the error vanishes if and only if the curvature is zero everywhere the iterative method generates term by term the asymptotic expansion of m e starting from m 0 the ith application of the algorithm yields the correct expansion coefficient at o e i while leaving the lower order coefficients invariant thus after l applications the expansion is accurate up to and including the terms of o e l the analytical results are illustrated on a planar system from enzyme kinetics michaelis menten henri and a model planar system due to davis and skodje

computation of nonautonomous invariant and inertial manifolds we derive a numerical scheme to compute invariant manifolds for time variant discrete dynamical systems i e nonautonomous difference equations our universally applicable method is based on a truncated lyapunov perron operator and computes invariant manifolds using a system of nonlinear algebraic equations which can be solved both locally using nonsmooth inexact newton and globally using continuation algorithms compared to other algorithms our approach is quite flexible since it captures time dependent nonsmooth noninvertible or implicit equations and enables us to tackle the full hierarchy of strongly stable stable and center stable manifolds as well as their unstable counterparts our results are illustrated using a test example and are applied to a population dynamical model and the henon map finally we discuss a linearly implicit euler bubnov galerkin discretization of a reaction diffusion equation in order to approximate its inertial manifold

global solutions of nonlinear hyperbolic equations for small initial data on considere des systemes quasilineaires d equations hyperboliques d ordre 2 qui sont des deformations non lineaires de l equation d onde on etudie l existence globale et le comportement asymptotique des solutions du probleme de cauchy quand la donnee initiale est suffisamment petite

uniform decay estimates and the lorentz invariance of the classical wave equation on etudie le comportement asymptotique des solutions de u 0 ou t 2 1 2 n 2 pour des conditions initiales u 0 u t g x en t 0 avec g reguliere a support compact dans r n

a parameterization method for the computation of invariant tori and their whiskers in quasi periodic maps rigorous results abstract in this paper we prove rigorous results on persistence of invariant tori and their whiskers the proofs are based on the parameterization method of x cabre e fontich r de la llave the parameterization method for invariant manifolds i manifolds associated to non resonant subspaces indiana univ math j 52 2 2003 283 328 x cabre e fontich r de la llave the parameterization method for invariant manifolds ii regularity with respect to parameters indiana univ math j 52 2 2003 329 360 the invariant manifolds results proved here include as particular cases of the usual strong stable and strong unstable manifolds but also include other non resonant manifolds the method lends itself to numerical implementations whose analysis and implementation is studied in a haro r de la llave a parameterization method for the computation of invariant tori and their whiskers in quasi periodic maps numerical algorithms preprint 2005 a haro r de la llave a parameterization method for the computation of invariant tori and their whiskers in quasi periodic maps numerical implementation and examples preprint 2005 the results are stated as a posteriori results namely that if one has an approximate solution which is not degenerate then one has a true solution not too far from the approximate one this can be used to validate the results of numerical computations

stabilization of the lattice boltzmann method by the h theorem a numerical test for a one dimensional benchmark shock tube problem we implement the lattice boltzmann method based on the h theorem i karlin a ferrante and h c ottinger europhys lett 47 182 1999 results of simulation demonstrate significant improvement of stability as compared to realizations without explicit entropic estimations

the universal limit in dynamics of dilute polymeric solutions the method of invariant manifold is developed for a derivation of reduced description in kinetic equations of dilute polymeric solutions it is demonstrated that this reduced description becomes universal in the limit of small deborah and weissenberg numbers and it is represented by the revised oldroyd 8 constants constitutive equation for the polymeric stress tensor coefficients of this constitutive equation are expressed in terms of the microscopic parameters a systematic procedure of corrections to the revised oldroyd 8 constants equations is developed results are tested with simple flows c 2000 elsevier science b v all rights reserved

grad s approximation for moving and stationary walls in entropic lattice boltzmann simulations we introduce a new generalized wall boundary condition for the entropic lattice boltzmann method elbm that is capable of handling complex flow geometries including moving walls the grad s ten moments approximation is employed to approximate the missing populations at the boundary nodes the new boundary condition significantly improves accuracy stability and sub grid features of lbm and in particular of its entropic variant elbm the proposed boundary condition is tested for a range of flows in two dimensions including planar channel flow couette flow flow past a circular cylinder and sedimenting circular particles further simulations involving complex geometries such as flapping wing and transversely oscillating cylinder demonstrate superior stability and accuracy of the new wall boundary condition

nonlinear observer design in the siegel domain we extend the method of kazantzis and kravaris systems control lett 34 1998 pp 241 247 for the design of an observer to a larger class of nonlinear systems the extended method is applicable to any real analytic observable nonlinear system it is based on the solution of a first order singular nonlinear pde this solution yields a change of state coordinates which linearizes the error dynamics under very general conditions the existence and uniqueness of the solution is proved lyapunov s auxiliary theorem and siegel s theorem are obtained as corollaries the technique is constructive and yields a method for constructing approximate solutions

isomorphism testing of boolean functions computable by constant depth circuits given two n variable boolean functions f and g we study the problem of computing an e approximate isomorphism between them an e approximate isomorphism is a permutation of the n boolean variables such that f x 1 x 2 x n and g x 1 x 2 x n differ on at most an e fraction of all boolean inputs 0 1 n we give a randomized 2 o n log n e o d time algorithm that computes an e approximate isomorphism between two isomorphic boolean functions f and g that are given by depth d circuits of poly n size where d is a constant independent of n for any positive e in contrast the best known algorithm for computing an exact isomorphism between n ary boolean functions has running time 2 o n 12 even for functions computed by poly n size dnf formulas our algorithm is based on a result for hypergraph isomorphism with bounded edge size 4 and the classical linial mansour nisan result on approximating small depth and size boolean circuits by small degree polynomials using fourier analysis 11