statistical theory for the stochastic burgers equation in the inviscid limit a statistical theory is developed for the stochastic burgers equation in the inviscid limit master equations for the probability density functions of velocity velocity difference and velocity gradient are derived no closure assumptions are made instead closure is achieved through a dimension reduction process namely the unclosed terms are expressed in terms of statistical quantities for the singular structures of the velocity field here the shocks master equations for the environment of the shocks are further expressed in terms of the statistics of singular structures on the shocks namely the points of shock generation and collisions the scaling laws of the structure functions are derived through the analysis of the master equations rigorous bounds on the decay of the tail probabilities for the velocity gradient are obtained using realizability constraints we also establish that the probability density function q of the velocity gradient decays as jj 7 2 as 1 c 2000 john wiley sons inc

symmetries of the turbulent state the emphasis of this review is on fundamental properties degree of universality and symmetries of the turbulent state the central questions are which symmetries remain broken even when the symmetry breaking factor reaches zero and which symmetries in contrast emerge in the state of developed turbulence we shall see that time reversibility is broken in all cases since turbulence is a far from equilibrium state accompanied by dissipation as far as scale invariance is concerned we argue that it is always broken in direct cascades toward small scales no matter how far one goes away from the pumping scale in contrast inverse cascades become scale invariant as they go toward large scales moreover some properties of the inverse cascades seem to be conformal invariant and related to schramm loewner evolution a class of random curves that can be mapped to a 1d brownian walk

mean field approximation and a small parameter in turbulence theory numerical and physical experiments on two dimensional 2d turbulence show that the differences of transverse components of velocity field are well described by gaussian statistics and kolmogorov scaling exponents in this case the dissipation fluctuations are irrelevant in the limit of small viscosity in general one can assume the existence of a critical space dimensionality d d c at which the energy flux and all odd order moments of velocity difference change sign and the dissipation fluctuations become dynamically unimportant at dl d c the flow can be described by the mean field theory leading to the observed gaussian statistics and kolmogorov scaling of transverse velocity differences it is shown that in the vicinity of d d c the ratio of the relaxation and translation characteristic times decreases to zero thus giving rise to a small parameter of the theory the expressions for pressure and dissipation contributions to the exact equation for the generating function of transverse velocity differences are derived in the vicinity of d d c the resulting equation describes experimental data on two dimensional turbulence and demonstrates the onset of intermittency as d ensuremath d c g0 and r stackrel ensuremath rightarrow l 0 in three dimensional flows in close agreement with experimental data in addition some exact relations between correlation functions of velocity differences are derived it is also predicted that the single point probability density function of transverse velocity components in developing as well as in the large scale stabilized two dimensional turbulence is a gaussian

structure of metastable states in the hopfield model an upper bound for the number of metastable states in the hopfield model is calculated as a function of the hamming fraction from an input pattern for all finite values of alpha the ratio of number of patterns to nodes the hamming fraction from the input pattern to the nearest metastable state is infinite when alpha 0 113 the bound also implies that there is a gap between a set of states close to the input pattern and another set centred around the hamming fraction 0 5 from it

statistical properties of four dimensional turbulence the energy transfer and small scale intermittency in decaying turbulence in four dimensions 4d are studied by direct numerical simulation and by spectral theory in comparison with three dimensions 3d the energy transfer is more efficient in 4d than in 3d hence the exponent of energy decay is larger the kolmogorov constant is 1 31 which is smaller than 1 72 in 3d the longitudinal third order structure function is confirmed to be governed by a 1 2 law 1 2 epsilonr instead of a 4 5 law in 3d the intermittency is weaker in 4d for the total dissipation rate nu summation operator i j partial differentialu j partial differentialx i 2 and the associated velocity difference spherical velocity difference on scale r than in 3d while it is slightly stronger for the surrogated dissipation rate nu partial differentialu 1 partial differentialx 1 2 and the associated longitudinal velocity difference the scaling exponents of the spherical and longitudinal velocity differences are also evaluated indicating that the spherical velocity difference is less intermittent in 4d while the longitudinal difference is more intermittent in 4d the distribution of the eigenvalues of the strain tensor is also examined it was also found that the normalized variance of the pressure gradient epsilon 3 2 nu 1 2 in 4d is smaller than in 3d the roles of the incompressibility condition the pressure gradient and the intermittency in d dimensional turbulence are examined and the importance of the longitudinal component of turbulent velocity field in the energy transfer toward small scales are discussed burgers turbulence as the asymptote of turbulence in large dimension is suggested

entropy estimate of a probability density from its mellin transform entropy estimate of heavy tailed probability distribution from its mellin transform is considered such an estimate is provided uniquely in terms of expected values obtained from mellin transform some particular cases are analytically investigated as a low number of expected values are available

singularities and the distribution of density in the burgers adhesion model abstract we are interested in the tail behavior of the pdf of mass density within the one and d dimensional burgers adhesion model used e g to model the formation of large scale structures in the universe after baryon photon decoupling we show that large densities are localized near kurtoparabolic singularities residing on space time manifolds of codimension two d 2 or higher d 3 for smooth initial conditions such singularities are obtained from the convex hull of the lagrangian potential the initial velocity potential minus a parabolic term the singularities contribute universal power law tails to the density pdf when the initial conditions are random in one dimension the singularities are preshocks nascent shocks whereas in two and three dimensions they persist in time and correspond to boundaries of shocks in all cases the corresponding density pdf has the exponent 7 2 originally proposed by e et al phys rev lett 78 1997 1904 for the pdf of velocity gradients in one dimensional forced burgers turbulence we also briefly consider models permitting particle crossings and thus multi stream solutions such as the zel dovich approximation and the jeans vlasov poisson equation with single stream initial data they have singularities of codimension one yielding power law tails with exponent 3

towards a dynamical theory of multifractals in turbulence abstract making use of the exact equations for structure functions supplemented by the equations for dissipative anomaly as well as an estimate for the lagrangian acceleration of fluid particles we obtain a main result of the multifractal theory of turbulence the central element of the theory is a dissipation cut off that depends on the order of the structure function an expression obtained for the exponents s n in the scaling relations u x n u x 2 n 2 re s n between the velocity gradients u x and the reynolds number re agrees well with experimental data

multiscale correlation functions in strong turbulence under the framework of yakhot phys rev e 57 1737 1998 we model intermittent structure functions in fully developed turbulence based on the experimentally supported markovian nature of turbulence cascades friedrich and peinke phys rev lett 78 863 1997 and calculate the multiscaling correlation functions fusion rules l vov and procaccia phys rev lett 76 2898 1996 which were experimentally tested benzi biferale and toschi phys rev lett 80 3244 1998 to be compatible with almost uncorrelated multiplicative process are analytically checked by direct calculations

exact solutions for a forced burgers equation with a linear external force abstract we investigate the solutions of the burgers equation t u x t d x 2 u x t x f x t u x t u x t x u x t x t where f x t is an external force and x t represents a forcing term this equation is first analyzed in the absence of the forcing term by taking f x t k 1 t k 2 t x into account for this case the solution obtained extends the usual one present in the ornstein uhlenbeck process and depending on the choice of k 1 t and k 2 t it can present a stationary state or an anomalous spreading afterwards the forcing terms x t 1 t 2 t x and x t 3 x 4 x 3 are incorporated in the previous analysis and exact solutions are obtained for both cases

renormalization group operator product expansion and anomalous scaling in models of turbulent advection recent progress on the anomalous scaling in models of turbulent heat and mass transport is reviewed with the emphasis on the approach based on the field theoretic renormalization group rg and operator product expansion ope in that approach the anomalous scaling is established as a consequence of the existence in the corresponding field theoretic models of an infinite number of dangerous composite fields operators with negative critical dimensions which are identified with the anomalous exponents this allows one to calculate the exponents in a systematic perturbation expansion similar to the e expansion in the theory of critical phenomena the rg and ope approach is presented in a self contained way for the example of a passive scalar field temperature concentration of an impurity etc advected by a self similar gaussian velocity ensemble with vanishing correlation time the so called kraichnan s rapid change model where the anomalous exponents are known up to order o e3 effects of anisotropy compressibility and the correlation time of the velocity field are discussed passive advection by non gaussian velocity field governed by the stochastic navier stokes equation and passively advected vector e g magnetic fields are considered

instanton theory of burgers shocks and intermittency a lagrangian approach to burgers turbulence is carried out along the lines of the field theoretical martin siggia rose formalism of stochastic hydrodynamics we derive from an analysis based on the hypothesis of unbroken galilean invariance the asymptotic form of the probability distribution function of negative velocity differences the origin of burgers intermittency is found to rely on the dynamical coupling between shocks identified to instantons and noncoherent background fluctuations which char22 then char22 cannot be discarded in a consistent statistical description of the flow

anomalous scaling in the n point functions of a passive scalar a recent analysis of the four point correlation function of the passive scalar advected by a time decorrelated random flow is extended to the n point case it is shown that all stationary state inertial range correlations are dominated by homogeneous zero modes of singular operators describing their evolution we compute analytically the zero modes governing the n point structure functions and the anomalous dimensions corresponding to them to the linear order in the scaling exponent of the two point function of the advecting velocity field the implications of these calculations for the dissipation correlations are discussed textcopyright 1996 the american physical society

exact resummations in the theory of hydrodynamic turbulence iii scenarios for anomalous scaling and intermittency elements of the analytic structure of anomalous scaling and intermittency in fully developed hydrodynamic turbulence are described we focus here on the structure functions of velocity differences that satisfy inertial range scaling laws s n r sim r zeta n and the correlation of energy dissipation k epsilon epsilon r sim r mu the goal is to understand the exponents zeta n and mu from first principles in paper ii of this series it was shown that the existence of an ultraviolet scale the dissipation scale eta is associated with a spectrum of anomalous exponents that characterize the ultraviolet divergences of correlations of gradient fields the leading scaling exponent in this family was denoted delta the exact resummation of ladder diagrams resulted in the calculation of delta which satisfies the scaling relation delta 2 zeta 2 in this paper we continue our analysis and show that nonperturbative effects may introduce multiscaling i e zeta n not being linear in n with the renormalization scale being the infrared outer scale of turbulence l it is shown that deviations from k41 scaling of s n r zeta n neq n 3 must appear if the correlation of dissipation is mixing i e mu 0 we derive an exact scaling relation mu 2 zeta 2 zeta 4 we present analytic expressions for zeta n for all n and discuss their relation to experimental data one surprising prediction is that the time decay constant tau n r propto r z n of s n r scales independently of n the dynamic scaling exponent z n is the same for all n order quantities z n zeta 2

universality classes for self similarity of noiseless multidimensional burgers turbulence and interface growth the present work is devoted to the evolution of random solutions of the unforced burgers and kpz equations in d dimensions in the limit of vanishing viscosity we consider a cellular model and as initial condition assign a value for the velocity potential chosen independently within each cell we show that the asymptotic behavior of the turbulence at large times is determined by the tail of the initial potential probability distribution function three classes of initial distribution leading to self similar evolution are identified a distributions with a power law tail b compactly supported potential c stretched exponential tails in class c we find that the mean potential mean height of the surface increases logarithmically with time and the turbulence energy e t mean square gradient of the surface decays as 1 t times a logarithmic correction in classes a and b we find that the changes in the mean potential and energy have a power law time dependence in class c the roughness of the surface measured by its mean square gradient may either decrease or increase with time we discuss also the influence of finite viscosity and long range correlation on the late stage evolution of the burgers turbulence

first principles derivation of static avalanche size distributions we study the energy minimization problem for an elastic interface in a random potential plus a quadratic well as the position of the well is varied the ground state undergoes jumps called shocks or static avalanches we introduce an efficient and systematic method to compute the statistics of avalanche sizes and manifold displacements the tree level calculation i e mean field limit is obtained by solving a saddle point equation graphically it can be interpreted as the sum of all tree graphs the 1 loop corrections are computed using results from the functional renormalization group at the upper critical dimension the shock statistics is described by the brownian force model bfm the static version of the so called alessandro beatrice bertotti montorsi abbm model in the nonequilibrium context of depinning this model can itself be treated exactly in any dimension and its shock statistics is that of a levy process contact is made with classical results in probability theory on the burgers equation with brownian initial conditions in particular we obtain a functional extension of an evolution equation introduced by carraro and duchon which recursively constructs the tree diagrams in the field theory

zero tension kardar parisi zhang equation in d 1 dimensions the joint probability distribution function pdf of the height and its gradients is derived for a zero tension d 1 dimensional kardar parisi zhang kpz equation it is proved that the height s pdf of zero tension kpz equation shows lack of positivity after a finite time t c the properties of zero tension kpz equation and its differences with the case that it possess an infinitesimal surface tension is discussed also potential relation between the time scale t c and the singularity time scale t c v 0 of the kpz equation with an infinitesimal surface tension is investigated

success or failure of a firm under different financing policies a dynamic stochastic model abstract the value of a firm evolves in time according to a standard ito s stochastic differential equation and under different types of financial support in particular both the cases of legal and illegal financing are considered the ruin success probabilities for the firm are evaluated in closed form via special functions in the various settings some functionals of the process which are relevant from the viewpoint of the financiers via kac s lemma are shown to obey a pde which is studied both with analytic tools second order odes of hypergeometric type and with numerical ones the case of insufficient payments is analyzed via stochastic calculus and a new model for loan renegotiation is offered

chaos and asymptotical stability in discrete time neural networks abstract this paper aims to theoretically prove by applying marotto s theorem that both transiently chaotic neural networks tcnn and discrete time recurrent neural networks drnn have chaotic structure a significant property tcnn and drnn is that they have only one bounded fixed point when absolute values of the self feedback connection weights in tcnn and the difference time in drnn are sufficiently large we show that this unique fixed point can actually evolve into a snap back repeller which generates chaotic structure if several conditions are satisfied on the other hand by using the lyapunov functions we also derive sufficient conditions on asymptotical stability for symmetrical versions of both tcnn and drnn under which tcnn and drnn asymptotically converge to a fixed point furthermore related bifurcations are also considered in this paper since both tcnn and drnn are not special but simple and general the obtained theoretical results hold for a wide class of discrete time neural networks to demonstrate the theoretical results of this paper better several numerical simulations are provided as illustrating examples

anomalous scaling nonlocality and anisotropy in a model of the passively advected vector field a model of the passive vector quantity advected by the gaussian velocity field with the covariance ensuremath propto ensuremath delta t ensuremath t ensuremath mathbf x ensuremath mathbf x ensuremath ensuremath varepsilon is studied the effects of pressure and large scale anisotropy are discussed the inertial range behavior of the pair correlation function is described by an infinite family of scaling exponents which satisfy exact transcendental equations derived explicitly in d dimensions by means of the functional techniques the exponents are organized in a hierarchical order according to their degree of anisotropy with the spectrum unbounded from above and the leading minimal exponent coming from the isotropic sector this picture extends to higher order correlation functions like in the scalar model the second order structure function appears nonanomalous and is described by the simple dimensional exponent s 2 ensuremath propto r 2 ensuremath ensuremath varepsilon for the higher order structure functions s 2n ensuremath propto r n 2 ensuremath ensuremath varepsilon ensuremath delta n the anomalous scaling behavior is established as a consequence of the existence in the corresponding operator product expansions of dangerous composite operators whose negative critical dimensions determine the anomalous exponents ensuremath delta n l0 a close formal resemblance of the model with the stirred navier stokes equation reveals itself in the mixing of relevant operators and is the main motivation of the paper using the renormalization group the anomalous exponents are calculated in the o ensuremath varepsilon approximation in large d dimensions for the even structure functions up to the twelfth order

stieltjes moment problem via fractional moments stieltjes moment problem is considered to recover a probability density function from the knowledge of its infinite sequence of ordinary moments the approximate density is obtained through maximum entropy technique under the constraint of few fractional moments the latter are numerically obtained from the infinite sequence of ordinary moments and are chosen in such a way as to convey the maximum information content carried by the ordinary moments as a consequence a model with few parameters is obtained and intrinsic numerical instability is avoided it is proved that the approximate density is useful for calculating expected values and some other interesting probabilistic quantities

associative memory based on parametrically coupled chaotic elements abstract in this paper we propose an associative memory system based on parametrically coupled chaotic elements the proposed system is obtained by adding a new parameter control to our previously proposed system a chaotic activity in an early association stage makes an efficient association over the memories that are stored by means of autocorrelational learning when the system successfully recalls the target memory the system s motion is dominated by a spatially coherent oscillation while unstable motions remain when the system fails to make the association in addition the system has a large memory capacity a comparison between the proposed system and an approach with a nonmonotonic transfer function is also shown

on the approximation of the stochastic burgers equation we prove mathematical approximation results for the hyperviscous burgers equation driven by additive gaussian noise in particular we show that solutions of approximating equations driven by a discretized noise converge towards the solution of the original equation when the discretization parameter gets small the convergence takes place in the expected value of arbitrary powers of certain norms i e all moments of the difference of the solutions tend to zero in certain function spaces for the hyperviscous burgers equation these results are applied to justify the approximation of certain correlation functions that play a major role in statistical turbulence theory

on the structure of correlation functions in the normal matrix model we study the structure of the normal matrix model nmm we show that all correlation functions of the model with an axially symmetric potential can be expressed in terms of a single holomorphic function of one variable this observation is used to demonstrate the exact solvability of the model the two point correlation function is calculated in the continuum limit the answer is proven to be universal i e potential independent up to a change of the scale in connection with nmm with a general polynomial potential we have developed a two dimensional free fermion formalism and constructed a family of completely integrable hierarchies of non linear differential equations which we call the extended kp n hierarchies the well known kp hierarchy is a lower dimensional reduction of this family the extended kp 1 hierarchy contains the 2 1 dimensional burgers equations the partition function of the n n nmm is the function of the extended kp n hierarchy which is invariant with respect to a subalgebra of an algebra of all infinitesimal diffeomorphisms of the plane

anomalous scaling of a passive scalar in the presence of strong anisotropy field theoretic renormalization group and the operator product expansion are applied to a model of a passive scalar quantity straight theta t x advected by the gaussian strongly anisotropic velocity field with the covariance infinity delta t t x x epsilon inertial range anomalous scaling behavior is established and explicit asymptotic expressions for the structure functions s n r identical with are obtained they are represented by superpositions of power laws the corresponding anomalous exponents which depend explicitly on the anisotropy parameters are calculated to the first order in epsilon in any space dimension d in the limit of vanishing anisotropy the exponents are associated with tensor composite operators built of the scalar gradients and exhibit a kind of hierarchy related to the degree of anisotropy the less is the rank the less is the dimension and consequently the more important is the contribution to the inertial range behavior the leading terms of the even odd structure functions are given by the scalar vector operators for the finite anisotropy the exponents cannot be associated with individual operators which are essentially mixed in renormalization but the aforementioned hierarchy survives for all the cases studied the second order structure function s2 is studied in more detail using the renormalization group and zero mode techniques the corresponding exponents and amplitudes are calculated within the perturbation theories in epsilon 1 d and in the anisotropy parameters if the anisotropy of the velocity is strong enough the skewness factor s 3 s 3 2 2 increases going down towards the depth of the inertial range the higher order odd ratios increase even if the anisotropy is weak

subensemble decomposition and markov process analysis of burgers turbulence a numerical and statistical study is performed to describe the positive and negative local subgrid energy fluxes in the one dimensional random force driven burgers turbulence burgulence we use a subensemble method to decompose the field into shock wave and rarefaction wave subensembles by group velocity difference we observe that the shock wave subensemble shows a strong intermittency which dominates the whole burgulence field while the rarefaction wave subensemble satisfies the kolmogorov 1941 k41 scaling law we calculate the two subensemble probabilities and find that in the inertial range they maintain scale invariance which is the important feature of turbulence self similarity we reveal that the interconversion of shock and rarefaction waves during the equation s evolution displays in accordance with a markov process which has a stationary transition probability matrix with the elements satisfying universal functions and when the time interval is much greater than the corresponding characteristic value exhibits the scale invariant property

pattern dynamics in spatiotemporal chaos pattern selection diffusion of defect and pattern competition intermettency abstract a class of coupled map lattices is investigated as a model for the spatiotemporal chaos as the parameters are changed phase transitions among the patterns are observed pattern selection through the chaotic motion of domain is observed chaos is suppressed by the selection selection of the zigzag pattern is studied where localized defects which separate the two anti phase domains are found they change chaotically in time and move randomly in space the diffusion of defect is related with its chaotic motion collapse of zigzag pattern and the defect turbulence are studied in connection with the crisis in high dimensional dynamical systems pattern selection with longer wavelengths is found in the stronger coupling regime transition from pattern selection phase to fully developed turbulence occurs via the intermittency selective flicker like noise and the pareto distribution of lifetime of pattern are observed at the intermittency regime for the quantitative characterization of each phase pattern distributions pattern transition matrix static and dynamical entropies lifetime distribution of pattern spatiotemporal power spectra and lyapunov spectra are calculated phase transition from the pattern selection to fully developed turbulence is investigated with the use of these quantifiers where some order parameters and critical indeces are introduced

two dimensional turbulence in the inverse cascade range numerical and physical experiments on forced two dimensional navier stokes equations show that transverse velocity differences are described by normal kolmogorov scaling proportional r 2n 3 and obey gaussian statistics since nontrivial scaling is a sign of the strong nonlinearity of the problem these two results seem to contradict each other a theory explaining these observations is presented in this paper the derived self consistent expression for the pressure gradient contributions leads to the conclusion that small scale transverse velocity differences are governed by a linear langevin like equation stirred by a nonlocal universal solution dependent gaussian random force this explains the experimentally observed gaussian statistics of transverse velocity differences and their kolmogorov scaling the solution for the pdf of longitudinal velocity differences is based on the numerical smallness of the energy flux in two dimensional turbulence the theory makes a few quantitative predictions that can be tested experimentally

real moments from moments and vice versa positive and negative real moments are calculated uniquely in terms of given moments when the underlying stieltjes moment problem is determinate as well under weak hypothese moments are calculated from real moments the approximate densities which allow us to calculate real moments and moments reciprocally are borrowed from maximum entropy or minimun directed divergence techniques

a generalization of scaling models of turbulence we report on some implications of the theory of turbulence developed by v yakhot phys rev e 57 2 1737 1998 in particular we focus on the expression for the scaling exponents n we show that yakhot s result contains three well known scaling models as special cases namely k41 k62 and the theory by v l vov and i procaccia phys rev e 62 6 8037 2000 the model furthermore yields a theoretical justification for the method of extended self similarity ess

circulation statistics in three dimensional turbulent flows we study the large ensuremath lambda limit of the loop dependent characteristic functional z ensuremath lambda mathrm exp i ensuremath lambda ensuremath oint c stackrel ensuremath rightarrow v ensuremath cdot d stackrel ensuremath rightarrow x related to the probability density function pdf of the circulation around a closed contour c the analysis is carried out in the framework of the martin siggia rose field theory formulation of the turbulence problem by means of the saddle point technique axisymmetric instantons labeled by the component ensuremath sigma mathrm zz of the strain field char22 a partially annealed variable in our formalism char22 are obtained for a circular loop in the x ensuremath y plane with radius defined in the inertial range fluctuations of the velocity field around the saddle point solutions are relevant leading to the lorentzian asymptotic behavior z ensuremath lambda ensuremath sim 1 ensuremath lambda 2 the o 1 ensuremath lambda 4 subleading correction and the asymmetry between right and left pdf tails due to parity breaking mechanisms are also investigated

turbulence properties and global regularity of a modified navier stokes equation abstract we introduce a modification of the navier stokes equation that has the remarkable property of possessing an infinite number of conserved quantities in the inviscid limit this new equation is studied numerically and turbulence properties are analyzed concerning energy spectra and scaling of structure functions the dissipative structures arising in this new equation are curled vortex sheets instead of the vortex tubes arising in navier stokes turbulence the numerically calculated scaling of structure functions is compared with a phenomenological model based on the she leveque approach finally for this equation we demonstrate global well posedness for sufficiently smooth initial conditions in the periodic case and in r 3 the key feature is the availability of an additional estimate which shows that the l 4 norm of the velocity field remains finite

moment information and entropy evaluation for probability densities abstract how much information does the sequence of integer moments carry about the corresponding unknown absolutely continuous distribution we prove that a reliable evaluation of the corresponding shannon entropy can be done by exploiting some known theoretical results on the entropy convergence uniquely involving exact moments without solving the underlying moment problem all the procedure essentially rests on the solution of linear systems with nearly singular matrices and hence it requires both calculations in high precision and a pre conditioning technique numerical examples are provided to support the theoretical results

fluctuating hydrodynamics and turbulence in a rotating fluid universal properties we analyze the statistical properties of three dimensional 3d turbulence in a rotating fluid to this end we introduce a generating functional to study the statistical properties of the velocity field mathbit v we obtain the master equation from the navier stokes equation in a rotating frame and thence a set of exact hierarchical equations for the velocity structure functions for arbitrary angular velocity mathbit ensuremath omega in particular we obtain the differential forms for the analogs of the well known von karman howarth relation for 3d fluid turbulence we examine their behavior in the limit of large rotation our results clearly suggest dissimilar statistical behavior and scaling along directions parallel and perpendicular to mathbit ensuremath omega the hierarchical relations yield strong evidence that the nature of the flows for large rotation is not identical to pure two dimensional flows to complement these results by using an effective model in the small ensuremath omega limit within a one loop approximation we show that the equal time correlation of the velocity components parallel to mathbit ensuremath omega displays kolmogorov scaling q ensuremath 5 3 wherein as for all other components the equal time correlators scale as q ensuremath 3 in the inertial range where mathbit q is a wave vector in 3d our results are generally testable in experiments and or direct numerical simulations of the navier stokes equation in a rotating frame

symmetries of the stochastic burgers equation all lie symmetries of the burgers equation driven by an external random force are found besides the generalized galilean transformations this equation is also invariant under the time reparametrizations it is shown that the gaussian distribution of a pumping force is not invariant under the symmetries and breaks them down leading to the non trivial vacuum instanton integration over the volume of the symmetry groups provides the description of fluctuations around the instanton and leads to an exactly solvable quantum mechanical problem

decaying turbulence in the generalised burgers equation we consider the generalised burgers equation frac partial u partial t f u frac partial u partial x nu frac partial 2 u partial x 2 0 t geqq 0 x in s 1 where f is strongly convex and is small and positive we obtain sharp estimates for sobolev norms of u upper and lower bounds differ only by a multiplicative constant then we obtain sharp estimates for the dissipation length scale and the small scale quantities which characterise the decaying burgers turbulence i e the structure functions and the energy spectrum the proof uses a quantitative version of an argument by aurell et al j fluid mech 238 467 486 1992 note that we are dealing with decaying as opposed to stationary turbulence thus our estimates are not uniform in time however they hold on a time interval t1 t2 where t1 and t2 depend only on f and the initial condition and do not depend on the viscosity these results allow us to obtain a rigorous theory of the one dimensional burgers turbulence in the spirit of kolmogorov s 1941 theory in particular we obtain two results which hold in the inertial range on one hand we explain the bifractal behaviour of the moments of increments or structure functions on the other hand we obtain an energy spectrum of the form k 2 these results remain valid in the inviscid limit

linearization of universal field equations the universal field equations recently constructed as examples of higher dimensional dynamical systems which admit an infinity of inequivalent lagrangians are shown to be linearised by a legendre transformation this establishes the conjecture that these equations describe integrable systems while this construction is implicit in general there exists a large class of solutions for which an explicit form may be written comment 11pp dtp 92 47 ni 92 011

numerical inversion of the mellin transform on the real line for heavy tailed probability density functions a method for numerical inversion on the real line of the mellin transform without reduction of the problem to the inversion of laplace transform is described maximum entropy technique is invoked in choosing the analytical form of the approximant function entropy convergence and then l 1 norm convergence is proved a stability analysis in evaluating entropy and expected values is illustrated an upper bound of the error in the expected values computation is provided in terms of entropy

on the evaluation of finite time ruin probabilities in a dependent risk model this paper establishes some enlightening connections between the explicit formulas of the finite time ruin probability obtained by ignatovand kaishev 2000 2004 and ignatov et al 2001 for a risk model allowing dependence the numerical properties of these formulas are investigated and efficient algorithms for computing ruin probability with prescribed accuracy are presented extensive numerical comparisons and examples are provided

intermittency of height fluctuations in stationary state of the kardar parisi zhang equation with infinitesimal surface tension in 1 1 dimensions the kardar parisi zhang kpz equation with infinitesimal surface tension dynamically develops sharply connected valley structures within which the height derivative is not continuous we discuss the intermittency issue in the problem of stationary state forced kpz equation in 1 1 dimensions it is proved that the moments of height increments c a behave as x 1 x 2 xi a with xi a a for length scales x 1 x 2 sigma the length scale sigma is the characteristic length of the forcing term we have checked the analytical results by direct numerical simulation

turbulence without pressure we develop exact field theoretic methods to treat turbulence when the effect of pressure is negligible we find explicit forms of certain probability distributions demonstrate that the breakdown of galilean invariance is responsible for intermittency and establish the operator product expansion we also indicate how the effects of pressure can be turned on perturbatively

implicit solutions of pdes further investigations of implicit solutions to nonlinear partial differential equations are pursued of particular interest are the equations which are lorentz invariant the question of which differential equations of second order for a single unknown are solved by the imposition of an inhomogeneous quadratic relationship among the independent variables whose coefficients are functions of is discussed and it is shown that if the discriminant of the quadratic vanishes then an implicit solution of the so called universal field equation is obtained the relation to the general solution is discussed

recovering a probability density function from its mellin transform the recovering of a probability density function f x from its mellin transform m s is considered the approximate f m x is chosen resorting to maximum entropy technique constrained by the first derivatives of m s evaluated at s 1 so the basic properties of a probability density are saved existence conditions of the approximate f m x entropy convergence and then l 1 norm convergence are proved some numerical examples are reported resorting to the mellin transform is an alternative to laplace one as the recovered probability distribution is heavy tailed or equivalently its probability density function has abscissa convergence laplace transform equal to 0

on the proximity of distributions in terms of coinciding fractional moments a bound on the absolute difference between two distributions based on entropy is introduced such a bound can be made as precise as desired according to the number of common fractional moments assumed the approximating distribution is obtained by maximizing the entropy constrained by fractional moments

statistical analysis and simulation of random shocks in stochastic burgers equation we study the statistical properties of random shock waves in stochastic burgers equation subject to random space time perturbations and random initial conditions by using the response excitation probability density function pdf method and the mori zwanzig mz formulation of irreversible statistical mechanics we derive exact reduced order equations for the one point and two point pdfs of the solution field in particular we compute the statistical properties of random shock waves in the inviscid limit by using an adaptive shock capturing discontinuous galerkin method in both physical and probability spaces we consider stochastic flows generated by high dimensional random initial conditions and random additive forcing terms yielding multiple interacting shock waves collapsing into clusters and settling down to a similarity state we also address the question of how random shock waves in space and time manifest themselves in probability space the proposed new mathematical framework can be applied to different conservation laws potentially leading to new insights into high dimensional stochastic dynamical systems and more efficient computational algorithms

chaotifying linear elman networks a linear model of recurrent neural networks called the elman networks is combined with the simple nonlinear modulo mod operation on its linear activated function so as to generate chaos purposely conditions on the weight matrix are obtained under which the generated chaos satisfies the mathematical definition of chaos in the sense of t y li and j a yorke 1975 some simple and representative weight matrices are constructed for designing such elman networks that can generate li yorke chaos several numerical simulations are shown to verify and visualize the design

structure functions in the stochastic burgers equation we study analytically and numerically structure functions s q r in the one dimensional burgers equation driven by noise with variance ensuremath propto k ensuremath beta in fourier space a when the noise is cut off at some length l c and b when it is not we present exact relations satisfied by s 3 r the von karman howarth relation and s 4 r that form the basis of our analysis when there is a cutoff length shocks occur and s q r ensuremath propto r for qg 2 for ensuremath delta lrl l c where ensuremath delta is the shock thickness for all ensuremath beta between 1 and 2 we deduce this behavior from the exact relations along with an ansatz that is verified numerically when there is no cutoff length multifractal behavior is known to occur only when ensuremath beta 0 through a study of exact expression s 3 we highlight the difference between multifractality in this case as compared to the case with a cutoff

risk model with fuzzy random individual claim amount in this paper we consider a risk model in which individual claim amount is assumed to be a fuzzy random variable and the claim number process is characterized as a poisson process the mean chance of the ultimate ruin is researched particularly the expressions of the mean chance of the ultimate ruin are obtained for zero initial surplus and arbitrary initial surplus if individual claim amount is an exponentially distributed fuzzy random variable the results obtained in this paper coincide with those in stochastic case when the fuzzy random variables degenerate to random variables finally two numerical examples are presented

decentralized formation control of mobile agents a unified framework abstract this paper studies the formation control problem for systems consisting of multiple mobile agents that are described by first order differential equations and second order differential equations respectively the following issues are investigated bounded input disturbance and time delay in derivation of the main results a blend of graph theoretic and system theoretic tools is employed where local potential functions and lasalle s invariant principle play central roles finally numerical examples which support the analytical results very well are also included

integrable generalizations of the two dimensional born infeld equation the born infeld equation in two dimensions is generalized to higher dimensions lorentz invariance is retained and the resulting system is completely integrable via linearization by legendre transformation this generalization retains homogeneity in second derivatives of the field

robust dynamic sliding mode control using adaptive renn for magnetic levitation system in this paper a robust dynamic sliding mode control system rdsmc using a recurrent elman neural network renn is proposed to control the position of a levitated object of a magnetic levitation system considering the uncertainties first a dynamic model of the magnetic levitation system is derived then a proportional integral derivative pid type sliding mode control system smc is adopted for tracking of the reference trajectories moreover a new pid type dynamic sliding mode control system dsmc is proposed to reduce the chattering phenomenon however due to the hardware being limited and the uncertainty bound being unknown of the switching function for the dsmc an rdsmc is proposed to improve the control performance and further increase the robustness of the magnetic levitation system in the rdsmc an renn estimator is used to estimate an unknown nonlinear function of lumped uncertainty online and replace the switching function in the hitting control of the dsmc directly the adaptive learning algorithms that trained the parameters of the renn online are derived using lyapunov stability theorem furthermore a robust compensator is proposed to confront the uncertainties including approximation error optimal parameter vectors and higher order terms in taylor series finally some experimental results of tracking the various periodic trajectories demonstrate the validity of the proposed rdsmc for practical applications

inverse cascade and intermittency of passive scalar in one dimensional smooth flow random advection of a lagrangian tracer scalar field ensuremath theta t x by a one dimensional spatially smooth and short correlated in time velocity field is considered scalar fluctuations are maintained by a source concentrated at the integral scale l the statistical properties of both scalar differences and the dissipation field are analytically determined exploiting the dynamical formulation of the model the gaussian statistics known to be present at small scales for incompressible velocity fields emerges here at large scales x ensuremath gg l these scales are shown to be excited by an inverse cascade of ensuremath theta 2 and the probability distribution function pdf of the corresponding scalar differences to approach the gaussian form as larger and larger scales are considered small scale x ensuremath ll l statistics is shown to be strongly non gaussian a collapse of scaling exponents for scalar structure functions takes place moments of order pg 1 all scale linearly independently of the order p smooth scaling x p is found for ensuremath 1lpl1 tails of the scalar difference pdf are exponential while at the center a cusped shape tends to develop when smaller and smaller ratios x l are considered the same tendency is present for the scalar gradient pdf with respect to the inverse of the p eclet number the pumping to diffusion scale ratio the tails of the latter pdf are however much more extended decaying as a stretched exponential of exponent 2 3 smaller than unity this slower decay is physically associated with the strong fluctuations of the dynamical dissipative scale

on finite time ruin probabilities in a generalized dual risk model with dependence abstract in this paper we study the finite time ruin probability in a reasonably generalized dual risk model where we assume any non negative non decreasing cumulative operational cost function and arbitrary capital gains arrival process establishing an enlightening link between this dual risk model and its corresponding insurance risk model explicit expressions for the finite time survival probability in the dual risk model are obtained under various general assumptions for the distribution of the capital gains in order to make the model more realistic and general different dependence structures among capital gains and inter arrival times and between both are also introduced and corresponding ruin probability expressions are also given the concept of alarm time as introduced in das and kratz 2012 is applied to the dual risk model within the context of risk capital allocation extensive numerical illustrations are provided

chaotification via system immersion by introducing a definition of partially chaotic for a kind of generalized chaotic system this paper discusses the chaotification problem with a new approach based on system immersion and manifold invariance the basic idea is to immerse the plant system into a reduced order chaotic system the proposed approach is also applied to chaotify systems with uncertain parameters illustrative examples with simulation results are presented to validate the proposed chaotification schemes

the truncated stieltjes moment problem solved by using kernel density functions in this work the problem of the approximate numerical determination of a semi infinite supported continuous probability density function pdf from a finite number of its moments is addressed the target space is carefully defined and an approximation theorem is proved establishing that the set of all convex superpositions of appropriate kernel density functions kdfs is dense in this space a solution algorithm is provided based on the established approximate representation of the target pdf and the exploitation of some theoretical results concerning moment sequence asymptotics the solution algorithm also permits us to recover the tail behavior of the target pdf and incorporate this information in our solution a parsimonious formulation of the proposed solution procedure based on a novel sequentially adaptive scheme is developed enabling a very efficient moment data inversion the whole methodology is fully illustrated by numerical examples

a note on proximity of distributions in terms of coinciding moments a bound on the absolute difference between two distributions based on entropy is introduced such a bound can be made as precise as desired according to the number of common moments assumed the approximating distribution is obtained by maximizing the entropy constrained by moments

burgers turbulence and random lagrangian systems we consider a spatially periodic inviscid random forced burgers equation in arbitrary dimension and the random time dependent lagrangian system related to it we construct a unique stationary distribution for viscosity solutions of the burgers equation we also show that with probability 1 there exists a unique minimizing trajectory for the random lagrangian system which generates a non trivial ergodic invariant measure for the non random skew product extension of the lagrangian system

mean field fluctuation of a network of chaotic elements remaining fluctuation and correlation in the large size limit fluctuation of the mean field is studied for a network of chaotic elements with the use of globally coupled maps as the size n is increased the variance of the mean field decreases up to a crossover size nc but then stays constant where the remaining variance is roughly proportional to 1nc the distribution of the mean field approaches a gaussian form up to nc but keeps some deviation for n nc the breakdown of the law of large numbers implies some hidden coherence among elements confirmed by finite mutual information and lyapunov spectra the breakdown is also supported by the instability of the self consistent solution for the perron frobenius operator the anomalous effect of additional noise to our system is also reported results from other globally coupled maps suggest that the law of large numbers is generally broken in a turbulent phase with local stretching and contraction but is satisfied for a map with stretching only

multi hamiltonian structure of the born infeld equation the multi hamiltonian structure conservation laws and higher order symmetries for the born infeld equation are exhibited a new transformation of the born infeld equation to the equations of a chaplygin gas is presented and explored the born infeld equation is distinguished among two dimensional hyperbolic systems by its wealth of such multi hamiltonian structures

intermittency exponents and energy spectrum of the burgers and kpz equations with correlated noise we numerically calculate the energy spectrum intermittency exponents and probability density p u 0 of the one dimensional burgers and kpz equations with correlated noise we have used pseudo spectral method for our analysis when of the noise variance of the burgers equation variance k 2 exceeds 3 2 large shocks appear in the velocity prole leading to hju k j 2 i k 2 and structure function hju x r t u x t j q i r suggesting that the burgers equation is intermittent for this range of for 1660 the prole is dominated by noise and the spectrum hjh k j 2 i of the corresponding kpz equation is in close agreement with the medina et al renormalization group predictions in the intermediate range 0 1 2 we have also calculated the energy cascade rates for all and found a constant ux for all 1 2 c 2000 published

numerical inversion of laplace transform on the real line from expected values a numerical method for inversion of laplace transform f s of probability densities given for real s 0 only is proposed fractional calculus allows us to have expected values from f s or f s then the maximum entropy technique provides the analytical form of the approximant fractional moments are mainly investigated entropy and cross entropy convergence are proved some numerical tests are illustrated

convergence of best entropy estimates given a finite number of moments of an unknown density on a finite measure space the best entropy estimate that nonnegative density x with the given moments which minimizes the boltzmann shannon entropy i x x log x is considered a direct proof is given that i has the kadec property in l if yn converges weakly to 37 and i yn converges to i 37 then yn converges to 37 in norm as a corollary it is obtained that as the number of given moments increases the best entropy estimates converge in li norm to the best entropy estimate of the limiting problem which is simply in the determined case furthermore for classical moment problems on intervals with strictly positive and sufficiently smooth error bounds and uniform convergence are actually obtained

taylor s frozen flow hypothesis in burgers turbulence by detailed analytical treatment of the shock dynamics in the burgers turbulence with large scale forcing we calculate the velocity structure functions between pairs of points displaced both in time and space our analytical treatment verifies the so called taylor s frozen flow hypothesis without relying on any closure and under very general assumptions we discuss the limitation of the hypothesis and show that it is valid up to time scales smaller than the correlation time scale of temporal velocity correlation function we support the analytical calculation by performing numerical simulation of the periodically kicked burgers equation

kinetic theoretical approach to turbulence in variable density incompressible statistically inhomogeneous fluids the kinetic theory is developed for the mass mixing of two incompressible immiscible fluids due to rayleigh taylor instability as an example for turbulence in variable density statistically inhomogeneous incompressible fluids an expression is derived for the fine grain force in terms of the mass density and velocity fields this expression enables the conversion of the navier stokes equation into an exact explicit conservation equation in phase space the equation is a generalization to the variable density case of the lundgren equation t s lundgren phys fluids 10 969 1967 the conserved quantity is the fine grain density velocity distribution fgdvd the fine grain mass density and fluid velocity fields are the two lowest moments of the fgdvd the joint density velocity probability density function dvpdf is the ensemble average of the fgdvd using detailed numerical solutions of the navier stokes equation it is found that the correlation between the acceleration and the fgdvd is weak this result identifies a small parameter which enables the derivation by controlled approximations of closed equations for the dvpdfs the lowest order yields the mean field approximation it is shown by a numerical solution of the closed kinetic equation in the mean field approximation that it properly describes the time evolution of the system for periods shorter than the relaxation time closure schemes beyond the mean field are discussed

extra dimensions and nonlinear equations solutions of nonlinear multi component euler monge partial differential equations are constructed in n spatial dimensions by dimension doubling a method that completely linearizes the problem nonlocal structures are an essential feature of the method the euler monge equations may be interpreted as a boundary theory arising from a linearized bulk system such that all boundary solutions follow from simple limits of those for the bulk

a nonhomogeneous risk model for insurance a discrete time risk model is proposed that describes the temporal evolution of the surplus of an insurance company at fixed dates the novelty of the model comes from its total nonhomogeneity in the sense that as in our previous paper 1 the premium income process is deterministic and nonuniform but in addition in the present work the successive claim amounts are independent and nonidentically distributed our purpose is to evaluate for this extended model the probability of ruin over any finite time horizon rather surprisingly the methodology which was developed in 1 can be generalized to the case of nonstationary claim amounts the key mathematical tool for that is a theory of pseudopolynomials of appell type furthermore it is shown that a similar approach can be applied to a multirisks version of the model as well as to a continuous time version

entropy estimate of probability densities having assigned moments stieltjes case we consider an absolutely continuous positive random variable having assigned the first algebraic moments in terms of given moments is provided an upper bound for entropy for all the distributions admitting moment generating function

probability densities in strong turbulence abstract in this work we using mellin s transform combined with the gaussian large scale boundary condition calculate probability densities pdfs of velocity increments p r u r velocity derivatives p u r and the pdf of the fluctuating dissipation scales q re where re is the large scale reynolds number the resulting expressions strongly deviate from the log normal pdf p l r u r often quoted in the literature it is shown that the probability density of the small scale velocity fluctuations includes information about the large integral scale dynamics which is responsible for the deviation of p r u r from p l r u r an expression for the function d h of the multifractal theory free from spurious logarithms recently discussed in u frisch m martins afonso a mazzino v yakhot j fluid mech 542 2005 97 is also obtained

role of pressure in turbulence there is very limited knowledge of the kinematical relations for the velocity structure functions higher than three instead the dynamical equations for the structure functions of the velocity increment are obtained from the navier stokes equation under the assumption of the local homogeneity and isotropy these equations contain the correlation between the velocity and pressure gradient increments which is very difficult to know theoretically and experimentally we have examined these dynamical relations by using direct numerical simulation data at very high resolution at large reynolds numbers and found that the contribution of the pressure term is important to the dynamics of the longitudinal velocity with large amplitudes the pressure term is examined from the view point of the conditional average and the role of the pressure term in the turbulence dynamics is discussed

three dimensional forced burgers turbulence supplemented with a continuity equation we investigate turbulent limit of the forced burgers equation supplemented with a continuity equation in three dimensions the scaling exponent of the conditional two point correlation function of density i e r x1 r x2 udu ux12x2u 2a 3 is calculated self consistently in the nonuniversal region from which we obtain a 353 also we derive an equation governing the evolution of the probability density function pdf of longitudinal velocity increments in length scale from which a possible mechanism for the dependence of the inertial pdf to one point urms is developed

maximum entropy in the finite stieltjes and hamburger moment problem a necessary and sufficient condition for the existence of the maximum entropy me function defined in an infinite or semiinfinite interval is provided the conclusions reached show that except in a few particular cases the necessary and sufficient conditions for the existence of maximum entropy function are identical to the conditions for the solution of the moment problem when the first m 1 moments are assigned even if the conclusions reached are very similar to the hausdorff case the specificity of the hamburger and stieltjes cases demands a different handling a sufficient condition for the entropy convergence of the resulting sequence of maximum entropy estimators to the entropy of the recovering function is also provided

entropy convergence in stieltjes and hamburger moment problem in the classical stieltjes and hamburger moment problem the sequence of maximum entropy approximants whose first m moments are equal to given ones is considered it is proved that whenever an infinite moment problem is determined then maximum entropy approximants converge in entropy to the function characterized by given moments entropy convergence is proved by using exclusively existence and uniqueness conditions