second order accurate projective integrators for multiscale problems we introduce new projective versions of second order accurate runge kutta and adams bashforth methods and demonstrate their use as outer integrators in solving stiff differential systems an important outcome is that the new outer integrators when combined with an inner telescopic projective integrator can result in fully explicit methods with adaptive outer step size selection and solution accuracy comparable to those obtained by implicit integrators if the stiff differential equations are not directly available our formulations and stability analysis are general enough to allow the combined outer inner projective integrators to be applied to legacy codes or perform a coarse grained time integration of microscopic systems to evolve macroscopic behavior for example

optimal design of magnitude responses of rational infinite impulse response filters this correspondence considers a design of magnitude responses of optimal rational infinite impulse response iir filters the design problem is formulated as an optimization problem in which a total weighted absolute error in the passband and stopband of the filters the error function reflects a ripple square magnitude is minimized subject to the specification on this weighted absolute error function defined in the corresponding passband and stopband as well as the stability condition since the cost function is nonsmooth and nonconvex while the constraints are continuous this kind of optimization problem is a nonsmooth nonconvex continuous functional constrained problem to address this issue our previous proposed constraint transcription method is applied to transform the continuous functional constraints to equality constraints then the nonsmooth problem is approximated by a sequence of smooth problems and solved via a hybrid global optimization method the solutions obtained from these smooth problems converge to the global optimal solution of the original optimization problem hence small transition bandwidth filters can be obtained

telescopic projective methods for parabolic differential equations projective methods were introduced in an earlier paper c w gear i g kevrekidis projective methods for stiff differential equations problems with gaps in their eigenvalue spectrum nec research institute report 2001 029 available from http www neci nj nec com homepages cwg projective pdf abbreviated version to appear in sisc as having potential for the efficient integration of problems with a large gap between two clusters in their eigenvalue spectrum one cluster containing eigenvalues corresponding to components that have already been damped in the numerical solution and one corresponding to components that are still active in this paper we introduce iterated projective methods that allow for explicit integration of stiff problems that have a large spread of eigenvalues with no gaps in their spectrum as arise in the semi discretization of pdes with parabolic components

global boundedness of moments of solutions of the boltzmann equation for forces of infinite range this paper proves the global boundedness in time of higher moments to a weak solution of the non linear space homogeneous boltzmann equation for inverse kth power forces with k 5 in the course of the proof a new collisional estimate of v s v s v s v s is obtained where v v are the velocities before and v v are the velocities after a binary collision

fluid dynamic limits of kinetic equations i formal derivations the connection between kinetic theory and the macroscopic equations of fluid dynamics is described in particular our results concerning the incompressible navier stokes equations are based on a formal derivation in which limiting moments are carefully balanced rather than on a classical expansion such as those of hilbert or chapman enskog the moment formalism shows that the limit leading to the incompressible navier stokes equations like that leading to the compressible euler equations is a natural one in kinetic theory and is contrasted with the systematics leading to the compressible navier stokes equations some indications of the validity of these limits are given more specifically the connection between the diperna lions renormalized solution of the classical boltzmann equation and the leray solution of the navier stokes equations is discussed

possibilities and limitations of gaussian closure approximations for phase ordering dynamics the nonlinear equations describing phase ordering dynamics can be closed by assuming the existence of an underlying gaussian stochastic field which is nonlinearly related to the observable order parameter field we discuss the relation between different implementations of the gaussian assumption and consider the limitations of this assumption for phase ordering dynamics the fact that the different approaches gives different results is a sign of the breakdown of the gaussian assumption we concentrate on the non conserved order parameter case but also touch on the conserved order parameter case we demonstrate that the gaussian assumption is fundamentally flawed in the latter case

constrained least squares design of 2 d fir filters we consider the design of 2 d linear phase finite impulse response fir filters according to the least squares ls error criterion subject to equality and or inequality constraints since we use a frequency domain formulation these constraints can be used to explicitly prescribe frequency dependent error tolerances the maximum minimum or fixed values of the frequency response at certain points and or regions our method combines lagrange multiplier and kuhn tucker theory to solve a much wider class of problems than do standard methods it allows arbitrary compromises between the ls and the equiripple design

the boltzmann equation for weakly inhomogeneous data we solve the initial value problem associated to the nonlinear boltzmann equation in the case in which the initial distribution has sufficiently small spatial gradients

fluctuation response relations for multitime correlations we show that time correlation functions of arbitrary order for any random variable in a statistical dynamical system can be calculated as higher order response functions of the mean history of the variable the response is to a control term added as a modification to the master equation for statistical distributions the proof of the relations is based upon a variational characterization of the generating functional of the time correlations the same fluctuation response relations are preserved within moment closures for the statistical dynamical system when these are constructed via the variational rayleigh ritz procedure for the two time correlations of the moment variables themselves the fluctuation response relation is equivalent to an onsager regression hypothesis for the small fluctuations for correlations of higher order there is a further effect in addition to such linear propagation of fluctuations present instantaneously the dynamical generation of correlations by nonlinear interaction of fluctuations in general we discuss some physical and mathematical aspects of the ans atze required for an accurate calculation of the time correlations we also comment briefly upon the computational use of these relations which is well suited for automatic differentiation tools an example will be given of a simple closure for turbulent energy decay which illustrates the numerical application of the relations

existence of monotone solutions of some difference equations with unstable equilibrium abstract we consider the difference equation x n 1 f x n x n 1 n 0 1 where the function f u v is continuous on i2 for some interval i 0 f 0 0 0 and f x x x for x i 0 assuming that f u c is decreasing in u on i and that f c v is increasing in v on i for any c i we establish a necessary and sufficient condition for existence of monotone solutions converging to the equilibrium x 0

domains of undefinable solutions of some equations and systems of difference equations for some classes of difference equations and systems of difference equations we find the sets of initial values for which solutions of these equations and systems are not well defined so called domains of undefinable solutions

on the symmetric form of systems of conservation laws with entropy abstract this paper reviews the symmetrizability of systems of conservation laws which possess entropy functions symmetric formulations in conservation form for the equations of gas dynamics are presented

linear stochastic models of nonlinear dynamical systems we investigate in this work the validity of linear stochastic models for nonlinear dynamical systems we exploit as our basic tool a previously proposed rayleigh ritz approximation for the effective action of nonlinear dynamical systems started from random initial conditions the present paper discusses only the case where the probability density function ansatz employed in the variational calculation is markovian i e is determined completely by the present values of the moment averages in this case we show that the rayleigh ritz effective action of the complete set of moment functions that are employed in the closure has a quadratic part which is always formally an onsager machlup action thus subject to satisfaction of the requisite realizability conditions on the noise covariance a linear langevin model will exist which reproduces exactly the joint two time correlations of the moment functions we compare our method with the closely related formalism of principal oscillation patterns pop which in the approach of penland is a method to derive such a linear langevin model empirically from time series data for the moment functions the predictive capability of the pop analysis compared with the rayleigh ritz result is limited to the regime of small fluctuations around the most probable future pattern finally we shall discuss a thermodynamics of statistical moments which should hold for all dynamical systems with stable invariant probability measures and which follows within the rayleigh ritz formalism

locating resonances for axiom a dynamical systems for a class of differentiable dynamical systems called axiom a systems it has been shown by pollicott and the author that correlation functions have fourier transforms which are meromorphic in a strip the poles or resonances are however not easy to locate this note reviews the results which are known and discusses a simple model where the position of resonances can be estimated the effect of noise is also discussed

analysis of the m1 model well posedness and diffusion asymptotics abstract this paper is devoted to the analysis of the m 1 model which arises in radiative transfer theory the derivation of the model is based on the entropy minimization principle which leads to a hyperbolic system of balance laws with relaxation in the multi dimensional case we establish the existence uniqueness of a globally defined smooth solution under a suitable smallness condition on the initial data in the one dimensional case we show that the smallness condition does not depend on the particles mean free path so that we can also rigorously justify the consistency of the model with the diffusion asymptotics the result extends the analysis of coulombel et al j f coulombel f golse t goudon diffusion approximation and entropy based moment closure for kinetic equations asymptotic analysis 45 2005 1 39 to the case where the entropy functional accounts for relaxation towards the planckian state which is physically more relevant

more on a rational second order difference equation abstract here we give natural explanations for some recent relations for the solutions of the following difference equation z n 2 z n 1 3 z n 2 2 z n 1 2 n n 0 and extend them for the case of complex initial values we also present some other properties related to solutions of the equation as well as some properties of the solutions of an associate difference equation of first order

dynamics of a rational difference equation in this paper we investigate the periodic character invariant intervals oscillation and global stability of all positive solutions of the equationx n 1 px n x n kq x n k n 0 1 where the parameters p and q and the initial conditions x k x 0 are nonnegative real numbers as might be expected the two cases p q give rise to different dynamic behaviors in particular our results solve the open problem proposed by kulenvic and ladas in their monograph dynamics of second order rational difference equations with open problems and conjectures chapman hall crc boca raton 2001

a weighted least squares method for the design of stable 1 d and 2 d iir digital filters we present a new approach to the least squares design of stable infinite impulse response iir digital filters the design is accomplished by using an iterative scheme in which the denominator polynomial obtained from the preceding iteration is treated as a part of the weighting function and each iteration is carried out by solving a standard quadratic programming problem that yields a stable rational function when the iteration converges a stable and truly least squares solution is obtained the method is then extended to address the least squares design of stable iir two dimensional 2 d filters examples are included to illustrate the proposed design techniques

a path integral formalism for non equilibrium hamiltonian statistical systems a path integral formalism for non equilibrium systems is proposed based on a manifold of quasi equilibrium densities a generalized boltzmann principle is used to weight manifold paths with the exponential of minus the information discrepancy of a particular manifold path with respect to full liouvillean evolution the likelihood of a manifold member at a particular time is termed a consistency distribution and is analogous to a quantum wavefunction the lagrangian here is of modified generalized onsager machlup form for large times and long slow timescales the thermodynamics is of ottinger form the proposed path integral has connections with those occuring in the quantum theory of a particle in an external electromagnetic field it is however entirely of a wiener form and so practical to compute finally it is shown that providing certain reasonable conditions are met then there exists a unique steady state consistency distribution

an adaptive notch filter with improved tracking properties an analysis of the properties of an adaptive notch filter anf applied to time varying frequency tracking is presented starting from the derivation of an expression for anf output power asymptotically optimal values for the pole contraction and forgetting factors are derived for recursive prediction error rpe type anf algorithms based on the obtained results a new anf algorithm that includes adaptation of both pole contraction and forgetting factors is proposed the given experimental results confirm the theoretical conclusions and show that the proposed algorithm is highly efficient in practice

peak constrained least squares optimization we presented the basic concepts for peak constrained least squares pcls optimization in previous papers we present advanced pcls optimization concepts in this paper this paper discusses only a few of the many ways that pcls design problems can be formulated we believe that the pcls optimality criterion can be customized to fit virtually any practical application for example pcls optimization can readily be used to design equalizers in the prefilter equalizer or interpolated fir implementations of filters

equation free implementation of statistical moment closures we present a general numerical scheme for the practical implementation of statistical moment closures suitable for modeling complex large scale nonlinear systems building on recently developed equation free methods this approach numerically integrates the closure dynamics the equations of which may not even be available in closed form although closure dynamics introduce statistical assumptions of unknown validity they can have significant computational advantages as they typically have fewer degrees of freedom and may be much less stiff than the original detailed model the numerical closure approach can in principle be applied to a wide class of nonlinear problems including strongly coupled systems either deterministic or stochastic for which there may be no scale separation we demonstrate the equation free approach for implementing entropy based eyink levermore closures on a nonlinear stochastic partial differential equation

hyperbolic conservation laws with stiff relaxation terms and entropy we study the limiting behavior of systems of hyperbolic conservation laws with stiff relaxation terms reduced systems inviscid and viscous local conservation laws and weakly nonlinear limits are derived through asymptotic expansions an entropy condition is introduced for n n systems that ensures the hyperbolicity of the reduced inviscid system the resulting characteristic speeds are shown to be interlaced with those of the original system moreover the first correction to the reduced system is shown to be dissipative a partial converse is proved for 2 2 systems this structure is then applied to study the convergence to the reduced dynamics for the 2 2 case 1994 john wiley sons inc

on the difference equation xn xn 2 bn cnxn 1xn 2 abstract we prove that the difference equation x n x n 2 b n c n x n 1 x n 2 n n 0 where b n n n 0 and c n n n 0 are sequences periodic with period two and the initial values x 2 x 1 are real numbers can be solved explicitly some applications of obtained formulae are given

renormalization group study of thea b diffusion limited reaction thea b diffusion limited reaction with equal initial densitiesa 0 b 0 n0 is studied by means of a field theoretic renormalization group formulation of the problem for dimensiond 2 an effective theory is derived from which the density and correlation functions call be calculated we find the density decays in time as a b sim c sqrt delta dt d 4 ford 4 with n0 c n 0 d 2 wherec is a universal constant andc is nonuniversal the calculation is extended to the case of unequal diffusion constantsd a d b resulting in a new amplitude but the same exponent ford 2 a controlled calculation is not possible but a heuristic argument is presented that the results above give at least the leading term in an e 2 d expansion finally we address reaction zones formed in the steady state by opposing currents ofa andb particles and derive scaling properties

self consistent formulations for stochastic nonlinear neuronal dynamics neural dynamics is often investigated with tools from bifurcation theory however many neuron models are stochastic mimicking fluctuations in the input from unknown parts of the brain or the spiking nature of signals noise changes the dynamics with respect to the deterministic model in particular classical bifurcation theory cannot be applied we formulate the stochastic neuron dynamics in the martin siggia rose de dominicis janssen msrdj formalism and present the fluctuation expansion of the effective action and the functional renormalization group frg as two systematic ways to incorporate corrections to the mean dynamics and time dependent statistics due to fluctuations in the presence of nonlinear neuronal gain to formulate self consistency equations we derive a fundamental link between the effective action in the onsager machlup om formalism which allows the study of phase transitions and the msrdj effective action which is computationally advantageous these results in particular allow the derivation of an om effective action for systems with non gaussian noise this approach naturally leads to effective deterministic equations for the first moment of the stochastic system they explain how nonlinearities and noise cooperate to produce memory effects moreover the msrdj formulation yields an effective linear system that has identical power spectra and linear response starting from the better known loopwise approximation we then discuss the use of the frg as a method to obtain self consistency beyond the mean we present a new efficient truncation scheme for the hierarchy of flow equations for the vertex functions by adapting the blaizot m endez and wschebor approximation from the derivative expansion to the vertex expansion the methods are presented by means of the simplest possible example of a stochastic differential equation that has generic features of neuronal dynamics

most probable histories for nonlinear dynamics tracking climate transitions the effective action provides an appropriate cost function to determine most probable or optimal histories for nonlinear dynamics with strong noise in such strong coupling problems a nonperturbative technique is required to calculate the effective action we have proposed a rayleigh ritz variational approximation which employs simple moment closures or intuitive guesses of the statistics to calculate the effective action we consider here an application to climate dynamics within a simple bimodal langevin model similar to that proposed by c nicolis and g nicolis tellus33 225 1981 capturing climate state transitions even in this simple model is known to present a serious problem for standard methods of data assimilation in contrast it is shown that the effective action for the climate history is already well approximated by a one moment closure and that the optimal minimizing history robustly tracks climate change even with large observation errors furthermore the hessian of the effective action provides the ensemble variance as a realistic measure of confidence level in the predicted optimal history

pcls iir digital filters with simultaneous frequency response magnitude and group delay specifications this paper presents the peak constrained least squares pcls approach to designing iir digital filters pcls iir digital filters that meet simultaneous specifications on the frequency response magnitude and the group delay are introduced as a point of reference we consider the iir digital filter design problem that appears in deczky s 1972 classic paper and in the popular textbook by oppenheim and schafer 1989 in addition the same design problem appears in the iir filter design chapter by higgins and munson 1993 in the handbook for digital signal processing by using our new algorithm with simultaneous optimization of the frequency response magnitude and the group delay we obtain a dramatic improvement in the solution of this classic iir digital filter design problem starting from the same filter structure and the same specifications for the frequency response magnitude as in the works of deczky oppenheim and schafer and higgins and munson we are able to reduce the group delay ripple by a factor of 35 in another design problem that originated in deczky s work we use pcls optimization to reduce the group delay ripple by a factor of 40 at the same time we reduce the stopband energy by 6 db without sacrificing any other performance measure the group delay ripple in this iir digital filter example is reduced to only spl plusmn 0 002 samples

design of fir filters in the complex domain the design of fir digital filters with a complex valued desired frequency response using the chebyshev error is investigated the complex approximation problem is converted into a real approximation problem which is nearly equivalent to the complex problem a standard linear programming algorithm for the chebyshev solution of overdetermined equations is used to solve the real approximation problem additional constraints are introduced which allow weighting of the phase and or group delay of the approximation digital filters are designed which have nearly constant group delay in the passbands the desired constant group delay which gives the minimum chebyshev error is found to be smaller than that of a linear phase filter of the same length these filters in addition to having a smaller approximately constant group delay have better magnitude characteristics than exactly linear phase filters with the same length the filters have nearly equiripple magnitude and group delay

fock space symbolic algebra and analytical solutions for small stochastic systems randomness is ubiquitous in nature from single molecule biochemical reactions to macroscale biological systems stochasticity permeates individual interactions and often regulates emergent properties of the system while such systems are regularly studied from a modeling viewpoint using stochastic simulation algorithms numerous potential analytical tools can be inherited from statistical and quantum physics replacing randomness due to quantum fluctuations with low copy number stochasticity nevertheless classical studies remained limited to the abstract level demonstrating a more general applicability and equivalence between systems in physics and biology rather than exploiting the physics tools to study biological systems here the fock space representation used in quantum mechanics is combined with the symbolic algebra of creation and annihilation operators to consider explicit solutions for the chemical master equations describing small well mixed biochemical or biological systems this is illustrated with an exact solution for a michaelis menten single enzyme interacting with limited substrate including a consideration of very short time scales which emphasizes when stiffness is present even for small copy numbers furthermore we present a general matrix representation for michaelis menten kinetics with an arbitrary number of enzymes and substrates that following diagonalization leads to the solution of this ubiquitous nonlinear enzyme kinetics problem for this a flexible symbolic maple code is provided demonstrating the prospective advantages of this framework compared to stochastic simulation algorithms this further highlights the possibilities for analytically based studies of stochastic systems in biology and chemistry using tools from theoretical quantum physics

global asymptotic stability of a second order rational difference equation abstract the main goal of this paper is to investigate the global asymptotic stability of the difference equation y n 1 p n y n p n y n 1 n 0 1 2 where p n if n is even if n is odd and 0 0 and the initial conditions y 1 y 0 0 we show that the unique equilibrium y 1 is globally asymptotically stable under the certain conditions

some aspects of an adaptive digital notch filter with constrained poles and zeros the paper derives the range of filter parameters which can be used for monitoring purposes in adaptive notch filtering applications the nonconvergence problem associated with the stochastic gauss newton algorithm is illustrated and the paper proposes the use of a less efficient recursive estimation algorithm the appropriate maximum likelihood method to overcome this problem results from simulations are presented for demonstration purposes

efficient numerical methods for radiation in gas turbines numerical methods for radiative heat transfer equations coupled to a temperature equation are considered efficient solution methods and approximate equations for this system are investigated and a comparative numerical study of the different approximations is given the approximate equations considered in this paper include moment methods and diffusive approximations fast iterative solvers for the problem like multilevel methods with suitable preconditioning are considered in detail numerical experiments and comparisons in different space dimensions and for various physical situations are presented

an efficient steady state solver for microflows with high order moment model in z hu r li and z qiao acceleration for microflow simulations of high order moment models by using lower order model correction j comput phys 327 225 244 2016 it has been successfully demonstrated that using lower order moment model correction is a promising idea to accelerate the steady state computation of high order moment models of the boltzmann equation to develop the existing solver the following aspects are studied in this paper first the finite volume method with linear reconstruction is employed for high resolution spatial discretization of the resulting equations second by introducing an appropriate parameter in the correction step it is found that the performance of the solver can be improved significantly i e more levels would be involved in the solver which further accelerates the convergence of the method third heun s method is employed as the smoother in each level to enhance the robustness of the solver numerical experiments in microflows are carried out to demonstrate the efficiency and to investigate the behavior of the new solver in addition several order reduction strategies for the choice of the order sequence of the solver are tested and the strategy ml 1 ml 2 is found to be most efficient

numerical schemes for hyperbolic conservation laws with stiff relaxation terms hyperbolic systems often have relaxation terms that give them a partially conservative form and that lead to a long time behavior governed by reduced systems that are parabolic in nature in this article it is shown by asymptotic analysis and numerical examples that semidiscrete high resolution methods for hyperbolic conservation laws fail to capture this asymptotic behavior unless the small relaxation rate is resolved by a fine spatial grid we introduce a modification of higher order godunov methods that possesses the correct asymptotic behavior allowing the use of coarse grids large cell peclet numbers the idea is to build into the numerical scheme the asymptotic balances that lead to this behavior numerical experiments on 2 2 systems verify our analysis

global asymptotic stability of a second order nonlinear difference equation in this paper the global asymptotic stability of the nonlinear difference equationx n 1 a bx na bx n cx n 1 n 0 1 is investigated where a b a b c 0 are real numbers and the initial conditions x 1 is nonnegative real numbers and x 0 is a positive real number we show that the unique positive equilibrium of the equation is globally asymptotically stable in particular our results solve two conjectures proposed by kulenovic and ladas in their monograph m r s kulenovic g ladas dynamics of second order rational difference equations with open problems and conjectures chapman hall crc boca raton 2002 and by kulenovic et al in their paper m r s kulenovic g ladas l f martins i w rodrigues the dynamics of x n 1 a bx na bx n cx n 1 facts and conjectures comput math appl 45 2003 1087 1099 099

a coupled system of rational difference equations abstract we investigate the global stability properties and asymptotic behavior of solutions of the recursive sequence where the parameters a b c and d are arbitrary positive numbers and the initial conditions x0 and y0 are arbitrary nonnegative numbers

kinetics of phase ordering in the o n model with a conserved order parameter we study the phase ordering dynamics of the o n model with a conserved order parameter for systems with topological defects we present results from both cell dynamical simulations and predictions of a gaussian auxiliary field gaf approximation for the xy n 2 model in two and three dimensions and the heisenberg n 3 model in three dimensions we describe the results for the structure factor s q and growth law l t from simulations the growth laws obtained are consistent with theoretical predictions based on energy scaling arguments the structure factor shows good dynamical scaling using a length extracted from its first moment the simulations are compared with the theoretical predictions of the gaf for the scaling functions our results show that the gaf gives a good qualitative description of most features of the structure factor however it overestimates the amplitude of the porod tail in the large q limit moreover for small q the structure factor exhibits a q2 behaviour instead of the expected generalized yeung result of q4

effective action for stochastic partial differential equations stochastic partial differential equations spdes are the basic tool for modeling systems where noise is important spdes are used for models of turbulence pattern formation and the structural development of the universe itself it is reasonably well known that certain spdes can be manipulated to be equivalent to nonquantum field theories that nevertheless exhibit deep and important relationships with quantum field theory in this paper we systematically extend these ideas we set up a functional integral formalism and demonstrate how to extract all the one loop physics for an arbitrary spde subject to arbitrary gaussian noise it is extremely important to realize that gaussian noise does not imply that the field variables undergo gaussian fluctuations and that these nonquantum field theories are fully interacting the limitation to one loop is not as serious as might be supposed experience with quantum field theories qfts has taught us that one loop physics is often quite adequate to give a good description of the salient issues the limitation to one loop does however offer marked technical advantages because at one loop almost any field theory can be rendered finite using zeta function technology we can sidestep the complications inherent in the martin siggia rose formalism the spde analog of the becchi rouet stora tyutin formalism used in qft and instead focus attention on a minimalist approach that uses only the physical fields this direct approach is the spde analog of canonical quantization using physical fields after setting up the general formalism for the characteristic functional partition function we show how to define the effective action to all loops and then focus on the one loop effective action and its specialization to constant fields the effective potential the physical interpretation of the effective action and effective potential for spdes is addressed and we show that key features carry over from qft to the case of spdes an important result is that the amplitude of the two point function governing the noise acts as the loop counting parameter and is the analog of planck s constant in this spde context we derive a general expression for the one loop effective potential of an arbitrary spde subject to translation invariant gaussian noise and compare this with the one loop potential for qft

eulerian field theoretic closure formalisms for fluid turbulence the formalisms of wyld ann phys 14 143 1961 and martin siggia and rose msr phys rev a 8 423 1973 address the closure problem of a statistical treatment of homogeneous isotropic turbulence hit based on techniques primarily developed for quantum field theory in the wyld formalism there is a well known double counting problem for which an ad hoc solution was suggested by lee ann phys 32 292 1965 we show how to implement this correction in a more natural way from the basic equations of the formalism this leads to what we call the improved wyld lee renormalized perturbation theory msr had noted that their formalism had more vertex functions than wyld s formalism and based on this felt wyld s formalism was incorrect however a careful comparison of both formalisms here shows that the wyld formalism follows a different procedure to that of the msr formalism and so the treatment of vertex corrections appears in different ways in the two formalisms taking that into account along with clarifications made to both formalisms we find that they are equivalent and we demonstrate this up to fourth order

moment closure hierarchies for kinetic theories this paper presents a systematicnonperturbative derivation of a hierarchy of closed systems of moment equations corresponding to any classical kinetic theory the first member of the hierarchy is the euler system which is based on maxwellian velocity distributions while the second member is based on nonisotropic gaussian velocity distributions the closure proceeds in two steps the first ensures that every member of the hierarchy is hyperbolic has an entropy and formally recovers the euler limit the second involves modifying the collisional terms so that members of the hierarchy beyound the second also recover the correct navier stokes behavior this is achieved through the introduction of a generalization of the bgk collision operator the simplest such system in three spatial dimensions is a 14 moment closure which also recovers the behavior of the grad 13 moment system when the velocity distributions lie near local maxwellians the closure procedure can be applied to a general class of kinetic theories

assessment of a high resolution centered scheme for the solution of hydrodynamical semiconductor equations hydrodynamical models are suitable to describe carrier transport in submicron semiconductor devices these models have the form of nonlinear systems of hyperbolic conservation laws with source terms coupled with poisson s equation in this article we examine the suitability of a high resolution centered numerical scheme for the solution of the hyperbolic part of these extended models in one space dimension because of the lack of physically significant exact analytical solutions the method is assessed against a benchmark for the system of compressible unsteady euler equations with source terms which has an exact solution the latter is shown to be nearly identical to the numerical one the method is then used to solve the extended hydrodynamical model em based on the maximum entropy closure recently introduced by anile romano and russo simulating a ballistic diode n n n which models a metal oxide semiconductor field effect transistor mosfet channel results are presented for the reduced and full equation em formulation at steady state for an initially discontinuous electron density at the junctions transient results show the evolution of highly nonlinear waves emanating from the neighborhood of the junctions

dem a variational treatment of dynamic systems this paper presents a variational treatment of dynamic models that furnishes time dependent conditional densities on the path or trajectory of a system s states and the time independent densities of its parameters these are obtained by maximising a variational action with respect to conditional densities under a fixed form assumption about their form the action or path integral of free energy represents a lower bound on the model s log evidence or marginal likelihood required for model selection and averaging this approach rests on formulating the optimisation dynamically in generalised coordinates of motion the resulting scheme can be used for online bayesian inversion of nonlinear dynamic causal models and is shown to outperform existing approaches such as kalman and particle filtering furthermore it provides for dual and triple inferences on a system s states parameters and hyperparameters using exactly the same principles we refer to this approach as dynamic expectation maximisation dem

the thermodynamic formalism for expanding maps letf x x be an expanding map of a compact space small distances are increased by a factor 1 a generating function z is defined which countsf periodic points with a weight one can express in terms of nonstandard fredholm determinants of certain transfer operators which can be studied by methods borrowed from statistical mechanics in this paper we review the spectral properties of the transfer operators and the corresponding analytic properties of z gibbs distributions and applications to julia sets are also discussed some new results are proved and some natural conjectures are proposed

global asymptotic behavior of a rational difference equation abstract the main goal of this paper is to investigate the global asymptotic behavior of the difference equation y n 1 y n 1 p n y n n 0 1 2 where p n is a period two sequence p n if n is even if n is odd with 0 and the initial conditions y 1 y 0 0 we obtain some global attractivity results for the equilibrium of this system for different values of the parameters

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

some applications of the method of moments for the homogeneous boltzmann and kac equations using the method of moments we prove that any polynomial moment of the solution of the homogeneous boltzmann equation with hard potentials or hard spheres is bounded provided that a moment of order strictly higher than 2 exists initially we also give partial results of convergence towards the maxwellian equilibrium in the case of soft potentials finally exponential as well as maxwellian estimates are introduced for the kac equation

action principle in nonequilibrium statistical dynamics we introduce a variational method for approximating distribution functions of dynamics with a liouville operator l in terms of a nonequilibriumaction functional for two independent left and right trial states the method is valid for deterministic or stochastic markov dynamics and for stationary or time dependent distributions a practical rayleigh ritz procedure is advanced whose inputs are a finitely parametrized ansatz for the trial states leading to a parametric action for their evolution the euler lagrange equations of the action principle are hamiltonian in form generally noncanonical this permits a simple identification of fixed points as critical points of the parametric hamiltonian we also establish a variational principle for low order statistics such as mean values and correlation functions by means of the least effective action the latter is a functional of the given variable which is positive and convex as a consequence of h older realizability inequalities its value measures the cost for a fluctuation from the average to occur and in a weak noise limit it reduces to the onsager machlup action in general the effective action is shown to arise from the nonequilibrium action functional by a constrained variation this result provides a rayleigh ritz scheme for calculating just the desired low order statistics with internal consistency checks less demanding than for the full distribution textcopyright 1996 the american physical society

stable iir notch filter design with optimal pole placement this paper presents a two stage approach for designing an infinite impulse response iir notch filter first the numerator of the transfer function of the iir notch filter is obtained by placing the zeros at the prescribed notch frequencies then the denominator polynomial is determined by using an iterative scheme in which the optimal pole placements are found by solving a standard quadratic programming problem for stability the pole radius in the single notch filter design is specified by the designer and in the multiple notch filter design the pole radius is constrained by using the implications of rouche s theorem examples are included to illustrate the effectiveness of the proposed techniques

an entropy stable discontinuous galerkin finite element moment method for the boltzmann equation this paper presents a numerical approximation technique for the boltzmann equation based on a moment system approximation in velocity dependence and a discontinuous galerkin finite element approximation in position dependence the closure relation for the moment systems derives from minimization of a suitable divergence this divergence based closure yields a hierarchy of tractable symmetric hyperbolic moment systems that retain the fundamental structural properties of the boltzmann equation the approximation in position dependence is based on the discontinuous galerkin finite element method the resulting combined discontinuous galerkin moment method corresponds to a galerkin approximation of the boltzmann equation in renormalized form the new momentclosure formulation engenders a new upwind numerical flux function based on half space integrals of the approximate distribution we establish that the proposed upwind flux ensures entropy dissipation of the discontinuous galerkin finite element approximation numerical results are presented for a one dimensional test case

on the recursive sequencexn 1 α xn 1 xn abstract we study the global stability the boundedness character and the periodic nature of the positive solutions of the difference equationxn 1 xn 1 xn where 0 and where the initial conditionsx 1andx0are arbitrary positive real numbers

the quantum liouville bgk equation and the moment problem abstract this work is devoted to the analysis of the quantum liouville bgk equation this equation arises in the work of degond and ringhofer on the derivation of quantum hydrodynamical models from first principles their theory consists in transposing to the quantum setting the closure strategy by entropy minimization used for kinetic equations the starting point is the quantum liouville bgk equation where the collision term is defined via a so called quantum local equilibrium defined as a minimizer of the quantum free energy under a local density constraint we then address three related problems we prove new results about the regularity of these quantum equilibria we prove that the quantum liouville bgk equation admits a classical solution and we investigate the long time behavior of the solutions the core of the proofs is based on a fine analysis of the properties of the minimizers of the free energy

maximisation of the entropy in non equilibrium this author proves that the maximisation of the entropy in non equilibrium is equivalent to the exploitation of the entropy inequality in extended thermodynamics

predictive turbulence modeling by variational closure we show that a variational implementation of probability density function pdf closures has the potential to make predictions of general turbulence mean statistics for which a priori knowledge of the incorrectness is possible this possibility exists because of realizability conditions on effective potential functions for general turbulence statistics these potentials measure the cost for fluctuations to occur away from the ensemble mean value in empirical time averages of the given variable and their existence is a consequence of a refined ergodic hypothesis for the governing dynamical system navier stokes dynamics approximations of the effective potentials can be calculated within pdf closures by an efficient rayleigh ritz algorithm the failure of realizability within a closure for the approximate potential of any chosen statistic implies a priori that the closure prediction for that statistic is not converged the systematic use of these novel realizability conditions within pdf closures is shown in a simple 3 mode system of lorenz to result in a statistically improved predictive ability in certain cases the variational method allows an a priori optimum choice of free parameters in the closure to be made

two langevin equations in the doi peliti formalism a system size expansion method is incorporated into the doi peliti formalism for stochastic chemical kinetics the basic idea of the incorporation is to introduce a new decomposition of unity associated with the so called cole hopf transformation this approach elucidates a relationship between two different langevin equations one is associated with a coherent state path integral expression and the other describes density fluctuations a simple reaction scheme is investigated as an illustrative example

fluctuations in the irreversible decay of turbulent energy a fluctuation law of the energy in freely decaying homogeneous and isotropic turbulence is derived within standard closure hypotheses for 3d incompressible flow in particular a fluctuation dissipation relation is derived which relates the strength of a stochastic backscatter term in the energy decay equation to the mean of the energy dissipation rate the theory is based on the so called effective action of the energy history and illustrates a rayleigh ritz method recently developed to evaluate the effective action approximately within probability density function pdf closures these effective actions generalize the onsager machlup action of nonequilibrium statistical mechanics to turbulent flow they yield detailed concrete predictions for fluctuations such as multi time correlation functions of arbitrary order which cannot be obtained by direct pdf methods they also characterize the mean histories by a variational principle