exact analysis of multihop amplify and forward relaying systems over general fading links a new theoretical method the generalized transformed characteristic function gtcf approach gives an exact integral solution for the probability density function of the end to end received signal to noise ratio in multihop amplify and forward relaying systems in turn the gtcf approach is used to obtain exact integral solutions for the ergodic capacity outage probability and average symbol error probability of multihop amplify and forward relaying systems the results are precise for any number of hops and can accommodate any channel fading distribution the theory provides the first exact theoretical solutions for numbers of hops larger than 2

minimum action method for the kardar parisi zhang equation we apply a numerical minimum action method derived from the wentzell freidlin theory of large deviations to the kardar parisi zhang equation for the height profile of a growing interface in one dimension we find that the transition pathway between different height configurations is determined by the nucleation and subsequent propagation of facets or steps corresponding to moving domain walls or growth modes in the underlying noise driven burgers equation this transition scenario is in accordance with recent analytical studies of the one dimensional kardar parisi zhang equation in the asymptotic weak noise limit we also briefly discuss transitions in two dimensions

self similar decay in the kraichnan model of a passive scalar we study the two point correlation function of a freely decaying scalar in kraichnan s model of advection by a gaussian random velocity field that is stationary and white noise in time but fractional brownian in space with roughness exponent 0 2 appropriate to the inertial convective range of the scalar we find all self similar solutions by transforming the scaling equation to kummer s equation it is shown that only those scaling solutions with scalar energy decay exponent a d 1 are statistically realizable where d is space dimension and 2 an infinite sequence of invariants jp p 0 1 2 is pointed out where j0 is corrsin s integral invariant but the higher invariants appear to be new we show that at least one of the invariants j0 or j1 must be nonzero possibly infinite for realizable initial data initial datum with a finite nonzero invariant the first being jp converges at long times to a scaling solution p with a d p p 0 1 the latter belongs to an exceptional series of self similar solutions with stretched exponential decay in space however the domain of attraction includes many initial data with power law decay when the initial datum has all invariants zero or infinite and also it exhibits power law decay then the solution converges at long times to a nonexceptional scaling solution with the same power law decay these results support a picture of a two scale decay with breakdown of self similarity for a range of exponents d a d 2 analogous to what has recently been found in the decay of burgers turbulence

dissipation statistics of a passive scalar in a multidimensional smooth flow we compute analytically the probability distribution function p e of the dissipation field e 2 of a passive scalar advected by a d dimensional random flow in the limit of large peclet and prandtl numbers batchelor kraichnan regime the tail of the distribution is a stretched exponential for e ln p e d2e 1 3

outage probability of diversity systems over generalized fading channels outage probability is an important performance measure of communication systems operating over fading channels relying on a simple and accurate algorithm for the numerical inversion of the laplace transforms of cumulative distribution functions we develop a moment generating function based numerical technique for the outage probability evaluation of maximal ratio combining mrc and postdetection equal gain combining egc in generalized fading channels for which the fading in each diversity path need not be independent identically distributed nor even distributed according to the same family of distributions the method is then extended to coherent egc but only for the case of nakagami m fading channels the mathematical formalism is illustrated by applying the method to some selected numerical examples of interest showing the impact of the power delay profile and the fading correlation on the outage probability of mrc and egc systems

generalized tunstall codes for sources with memory tunstall codes are variable to fixed length codes that maximize the expected number of source letters per dictionary string for discrete memoryless sources we analyze a generalization of tunstall coding to sources with memory and demonstrate that as the dictionary size increases the number of code letters per source symbol comes arbitrarily close to the minimum among all variable to fixed length codes of the same size we also find the asymptotic relationship between the dictionary size and the average length of a dictionary entry

large order asymptotes of the quantum field expansions for the kraichnan model of passive scalar advection a simple model of a passive scalar quantity advected by a gaussian non solenoidal compressible velocity field is considered large order asymptotes of quantum field expansions are investigated by the instanton approach the existence of a finite convergence radius of the series is proved the type and the position of the singularity of the series in a regularization parameter e are determined anomalous exponents of the main contributions to the structural functions are resummed using new information about the series convergence and two known orders of the e expansion

downlink coverage analysis for a finite 3 d wireless network of unmanned aerial vehicles in this paper we consider a finite network of unmanned aerial vehicles serving a given region modeling this network as a uniform binomial point process we derive the downlink coverage probability of a reference receiver located at an arbitrary position on the ground assuming nakagami m fading for all wireless links the reference receiver is assumed to connect to its closest transmitting node as is usually the case in cellular systems after deriving the distribution of distances from the reference receiver to the serving and interfering nodes we derive an exact expression for downlink coverage probability in terms of the derivative of laplace transform of interference power distribution in the downlink of this system it is not unusual to encounter scenarios in which the line of sight component is significantly stronger than the reflected multipath components to emulate such scenarios we also derive the coverage probability in the absence of fading from the results of nakagami m fading by taking the limit m to infty using asymptotic expansion of incomplete gamma function we concretely show that this limit reduces to a redundant condition consequently we derive an accurate coverage probability approximation for this case using dominant interferer based approach in which the effect of dominant interferer is exactly captured and the residual interference from other interferers is carefully approximated we then derive the bounds of the approximate coverage probability using berry esseen theorem our analyses reveal several useful trends in coverage probability as a function of height of the transmitting nodes and the location of reference receiver on the ground

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

instantons in the burgers equation the instanton solution for the forced burgers equation is found this solution describes the exponential tail of the probability distribution function of velocity differences in the region where shock waves are absent that is for large positive velocity differences the results agree with the one found recently by polyakov who used the operator product conjecture if this conjecture is true then our wkb asymptotics of the wyld functional integral should be exact to all orders of perturbation expansion around the instanton solution we also generalized our solution for the arbitrary dimension of the burgers kpz equation as a result we found the asymptotics of the angular dependence of the velocity difference probability distribution function textcopyright 1996 the american physical society

spectral compact difference hybrid computation of passive scalar in isotropic turbulence an accurate and efficient hybrid numerical method is developed for direct numerical simulation of passive scalar in homogeneous turbulence for the schmidt number 1 and 50 the hybrid method uses the standard fourier spectral method for the incompressible navier stokes equation and the combined compact difference scheme for the passive scalar transport equation accuracy of the method is carefully examined by comparing with the full spectral method regarding the spectra probability density function field structure of the passive scalar and is found to be very satisfactory the computational time for the hybrid method is decreased by 26 for the schmidt number 1 when compared to the full spectral method and by 77 for the schmidt number 50 when the number of grid points for the velocity field is reduced under the scale separation

kolmogorov turbulence in a random force driven burgers equation anomalous scaling and probability density functions high resolution numerical experiments described in this work show that velocity fluctuations governed by the one dimensional burgers equation driven by a white in time random noise with the spectrum ensuremath vert f k mathrm ensuremath vert 2 ifmmode bar else textasciimacron fi ensuremath propto mathit k mathrm ensuremath 1 exhibit a biscaling behavior all moments of velocity differences mathit s mathit n mathrm ensuremath le 3 r ensuremath vert u x r u x mathrm ensuremath vert mathit n ifmmode bar else textasciimacron fi ensuremath equiv ensuremath vert ensuremath delta u mathrm ensuremath vert mathit n ifmmode bar else textasciimacron fi ensuremath propto mathit r mathit n 3 while mathit s mathit n g3 r ensuremath propto mathit r mathit n ensuremath xi with ensuremath xi mathit n ensuremath approxeq 1 for real ng0 chekhlov and yakhot phys rev e 51 r2739 1995 the probability density function which is dominated by coherent shocks in the interval ensuremath delta u0 is scrp ensuremath delta u r ensuremath propto ensuremath delta u mathrm ensuremath mathit q with q ensuremath approxeq 4 a phenomenological theory describing the experimental findings is presented

non gaussian invariant measures for the majda model of decaying turbulent transport the problem of turbulent transport of a scalar field by a random velocity field is considered the scalar field amplitude exhibits rare but very large fluctuations whose typical signature is fatter than gaussian tails for the probability distribution of the scalar the existence of such large fluctuations is related to clustering phenomena of the lagrangian paths within the flow this suggests an approach to turn the large deviation problem for the scalar field into a small deviation or small ball problem for some appropriately defined process measuring the spreading with time of the lagrangian paths here such a methodology is applied to a model proposed by majda consisting of a white in time linear shear flow and some generalizations of it where the velocity field has finite or even infinite correlation time the non gaussian invariant measure for the reduced scalar field is derived and in particular it is shown that the one point distribution of the scalar has stretched exponential tails with a stretching exponent depending on the parameters in the model different universality classes for the scalar behavior are identified which all other parameters being kept fixed display a one to one correspondence with an exponent measuring time persistence effects in the velocity field c 2001 john wiley sons inc

structure of the three point correlation function of a passive scalar in the presence of a mean gradient the three point correlation function of a passive scalar advected by a random incompressible velocity field in the presence of a mean gradient is investigated by means of phenomenological hopf equations numerical solutions are provided in the case where the velocity is gaussian white in time and with a power law in space v stackrel ensuremath rightarrow r ensuremath v stackrel ensuremath rightarrow 0 2 ensuremath sim r 2 ensuremath ensuremath epsilon and in the model introduced by shraiman and siggia c r acad sci 321 279 1995 anomalous scaling exponents are found in both models the numerics agrees with all the available analytic perturbative results in addition the angular dependence of the correlation function is explicitly determined in the batchelor limit of random advection by a smooth velocity field the exponent is found to remain very close to 1 as found experimentally in this limit the three point correlation function is found to be very well represented away from collinearity by an explicit integral representation

instanton filtering for the stochastic burgers equation we address the question of whether one can identify instantons in direct numerical simulations of the stochastically driven burgers equation for this purpose we first solve the instanton equations using the chernykh stepanov method 2001 phys rev e 64 026306 these results are then compared to direct numerical simulations by introducing a filtering technique to extract prescribed rare events from massive data sets of realizations using this approach we can extract the entire time history of the instanton evolution which allows us to identify the different phases predicted by the direct method of chernykh and stepanov with remarkable agreement

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

particle dispersion in a multidimensional random flow with arbitrary temporal correlations abstract we study the statistics of relative distances r t between fluid particles in a spatially smooth random flow with arbitrary temporal correlations using the space dimensionality d as a large parameter we develop an effective description of lagrangian dispersion we describe the exponential growth of relative distances r 2 t exp 2 t at different values of the ratio between the correlation and turnover times we find the stretching correlation time which determines the dependence of r 1 r 2 on the difference t1 t2 the calculation of the next cumulant of r2 shows that statistics of r2 is nearly gaussian at small times as long as d 1 and becomes log normal at large times when large d approach fails for high order moments the crossover time between the regimes is the stretching correlation time which surprisingly appears to depend on the details of the velocity statistics at t we establish the dispersion of the ln r2 in the log normal statistics

transmission capacity of wireless ad hoc networks with outage constraints in this paper upper and lower bounds on the transmission capacity of spread spectrum ss wireless ad hoc networks are derived we define transmission capacity as the product of the maximum density of successful transmissions multiplied by their data rate given an outage constraint assuming that the nodes are randomly distributed in space according to a poisson point process we derive upper and lower bounds for frequency hopping fh cdma and direct sequence ds cdma ss networks which incorporate traditional modulation types no spreading as a special case these bounds cleanly summarize how ad hoc network capacity is affected by the outage probability spreading factor transmission power target signal to noise ratio snr and other system parameters using these bounds it can be shown that fh cdma obtains a higher transmission capacity than ds cdma on the order of m sup 1 2 spl alpha where m is the spreading factor and spl alpha 2 is the path loss exponent a tangential contribution is an apparently novel technique for obtaining tight bounds on tail probabilities of additive functionals of homogeneous poisson point processes

exact results for conditional means of a passive scalar in certain statistically homogeneous flows for a passive scalar t r t randomly advected by a statistically homogeneous flow the probability density function pdf of its fluctuation can in general be expressed in terms of two conditional means 2 t t and t 2 t we find that in some special cases either one of the two conditional means can be obtained explicitly from the equation of motion in the cases when there is no external source and that the normalized fluctuation reaches a steady state or when a steady state results from a negative damping 2 t t t 2 t 2 t the linearity of the conditional mean in these cases follows directly from the fact that the advection equation of a passive scalar is linear on the other hand when the scalar is supported by a homogeneous white in time external source t 2 t t 2

two different rapid decorrelation in time limits for turbulent diffusion a turbulent diffusion model in which the velocity field is gaussian and rapidly decorrelating in time grdt has been widely used recently in an endeavor to understand the emergence of anomalous scaling behavior of physical fields in fluid mechanics from the underlying stochastic partial differential equations the utility of the grdt model is the fact that correlation functions of the passive scalar field solve closed partial differential equations the usual moment closure obstacle is averted we study here the sense in which the grdt model describes turbulent diffusion by a general non gaussian velocity field with nontrivial temporal structure in the limit in which the correlation time of the velocity field is taken to zero when the velocity field is rescaled in a particular manner in this rapid decorrelation limit then a limit theorem of khas minskii indeed shows that the passive scalar statistics are described asymptotically by the grdt model for a broad class of velocity field models we provide however an explicit example of a poisson blob model velocity field which has two different well defined rapid decorrelation in time limits in one the passive scalar correlation functions converge to those of the grdt model and in the other they converge to a distinct nontrivial limit in which the correlation functions do not solve closed pde s we provide both mathematical and heuristic explanations for the differences between these two limits the conclusion is that the grdt model provides a universal description of the rapid decorrelation in time limit of general non gaussian velocity field models only when the velocity field is rescaled in a particular manner during the limit process

persistent stability of a chaotic system abstract we report that trajectories of a one dimensional model for inertial particles in a random velocity field can remain stable for a surprisingly long time despite the fact that the system is chaotic we provide a detailed quantitative description of this effect by developing the large deviation theory for fluctuations of the finite time lyapunov exponent of this system specifically the determination of the entropy function for the distribution reduces to the analysis of a schrodinger equation which is tackled by semi classical methods the system has generic instability properties and we consider the broader implications of our observation of long term stability in chaotic systems

large negative velocity gradients in burgers turbulence we consider one dimensional burgers equation driven by large scale white in time random force the tails of the velocity gradients probability distribution function pdf are analyzed by saddle point approximation in the path integral describing the velocity statistics the structure of the saddle point instanton that is the velocity field configuration realizing the maximum of probability is studied numerically in details the numerical results allow us to find analytical solution for the long time part of the instanton its careful analysis confirms the result of balkovsky et al phys rev lett 78 1452 1997 based on short time estimations that the left tail of pdf has the form mathrm ln mathcal p u x ensuremath propto ensuremath u x 3 2

anomalous scaling for a passive scalar near the batchelor limit a class of phenomenological hopf equations describing mixing of a passive scalar by random flow close to the batchelor limit i e advection by random strain and vorticity is analyzed in the batchelor limit multipoint correlators of the scalar are constructed explicitly by exploiting the sl n r symmetry of the hopf operator hopf equations close to this integrable limit are solved via singular perturbation theory based on matched asymptotic expansions the solution for the three point correlator exhibits anomalous scaling indicating persistence of the small scale anisotropy for the scalar in addition to the exponent the full configuration dependence of the correlator is obtained s1063 651x 98 08703 0 pacs number s 47 27 2i

stochastic least action principle for the incompressible navier stokes equation abstract we formulate a stochastic least action principle for solutions of the incompressible navier stokes equation which formally reduces to hamilton s principle for the incompressible euler solutions in the case of zero viscosity we use this principle to give a new derivation of a stochastic kelvin theorem for the navier stokes equation recently established by constantin and iyer which shows that this stochastic conservation law arises from particle relabelling symmetry of the action we discuss issues of irreversibility energy dissipation and the inviscid limit of navier stokes solutions in the framework of the stochastic variational principle in particular we discuss the connection of the stochastic kelvin theorem with our previous martingale hypothesis for fluid circulations in turbulent solutions of the incompressible euler equations

instanton for the kraichnan passive scalar problem we consider high order correlation functions of the passive scalar in the kraichnan model using the instanton formalism we find the scaling exponents ensuremath zeta n of the structure functions s n for n ensuremath gg 1 under the additional condition d ensuremath zeta 2 ensuremath gg 1 where d is the dimensionality of space at nl n c where n c d ensuremath zeta 2 2 2 ensuremath ensuremath zeta 2 the exponents are ensuremath zeta n ensuremath zeta 2 4 2n ensuremath n 2 n c while at ng n c they are n independent ensuremath zeta n ensuremath zeta 2 n c 4 we also estimate n dependent factors in s n particularly their behavior at n close to n c

precise minimax redundancy and regret recent years have seen a resurgence of interest in redundancy of lossless coding the redundancy regret of universal fixed to variable length coding for a class of sources determines by how much the actual code length exceeds the optimal ideal over the class code length in a minimax scenario one finds the best code for the worst source either in the worst case called also maximal minimax or on average we first study the worst case minimax redundancy over a class of stationary ergodic sources and replace shtarkov s bound by an exact formula among others we prove that a generalized shannon code minimizes the worst case redundancy derive asymptotically its redundancy and establish some general properties this allows us to obtain precise redundancy for memoryless markov and renewal sources for example we present the exact constant of the redundancy for memoryless and markov sources by showing that the integer nature of coding contributes log logm m 1 logm o 1 where m is the size of the alphabet then we deal with the average minimax redundancy and regret our approach here is orthogonal to most recent research in this area since we aspire to show that asymptotically the average minimax redundancy is equivalent to the worst case minimax redundancy for some classes of sources after formulating some general bounds relating these two redundancies we prove our assertion for memoryless and markov sources nevertheless we provide evidence that maximal redundancy of renewal processes does not have the same leading term as the average minimax redundancy however our general results show that maximal and average regrets are asymptotically equivalent

relations between redundancy patterns of the shannon code and wave diffraction patterns of partially disordered media the average redundancy of the shannon code rn as a function of the block length n is known to exhibit two very different types of behavior depending on the rationality or irrationality of certain parameters of the source it either converges to 1 2 as n grows without bound or it may have a nonvanishing oscillatory quasi periodic pattern around the value 1 2 for all large n in this paper we make an attempt to shed some insight into this erratic behavior of rn by drawing an analogy with the realm of physics of wave propagation in particular the elementary theory of scattering and diffraction it turns out that there are two types of behavior of wave diffraction patterns formed by crystals which are correspondingly analogous to the two types of patterns of rn when the crystal is perfect the diffraction intensity spectrum exhibits very sharp peaks a k a bragg peaks at wavelengths of full constructive interference these wavelengths correspond to the frequencies of the harmonic waves of the oscillatory mode of rn on the other hand when the crystal is imperfect and there is a considerable degree of disorder in its structure the bragg peaks disappear and the behavior of this mode is analogous to the one where rn is convergent

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

a path integration formulation of stochastic lagrangian models of turbulent flow abstract in analogy with the feynman kac path integration formulation of quantum mechanics a general path integration formulation of stochastic lagrangian models of turbulent flow is developed with respect to the equivalence with the usual eulerian and stochastic lagrangian descriptions we then discuss efficient numerical methods for the realisation of the lagrangian solutions via path integration which provide an effective alternative to the simpler monte carlo approach

non hermitian localization and population biology the time evolution of spatial fluctuations in inhomogeneous d dimensional biological systems is analyzed a single species continuous growth model in which the population disperses via diffusion and convection is considered time independent environmental heterogeneities such as a random distribution of nutrients or sunlight are modeled by quenched disorder in the growth rate linearization of this model of population dynamics shows that the fastest growing localized state dominates in a time proportional to a power of the logarithm of the system size using an analogy with a schr odinger equation subject to a constant imaginary vector potential we propose a delocalization transition for the steady state of the nonlinear problem at a critical convection threshold separating localized and extended states in the limit of high convection velocity the linearized growth problem in d dimensions exhibits singular scaling behavior described by a d ensuremath 1 dimensional generalization of the noisy burgers equation with universal singularities in the density of states associated with disorder averaged eigenvalues near the band edge in the complex plane the burgers mapping leads to unusual transverse spreading of convecting delocalized populations

on the average redundancy rate of the lempel ziv code in this paper we settle a long standing open problem concerning the average redundancy r sub n of the lempel ziv 78 lz78 code we prove that for a memoryless source the average redundancy rate attains asymptotically er sub n a spl delta n log n o log log n log sup 2 n where a is an explicitly given constant that depends on source characteristics and spl delta x is a fluctuating function with a small amplitude we also derive the leading term for the kth moment of the number of phrases we conclude by conjecturing a precise formula on the expected redundancy for a markovian source the main result of this paper is a consequence of the second order properties of the lempel ziv algorithm obtained by jacquet and szpankowski 1995 these findings have been established by analytical techniques of the precise analysis of algorithms we give a brief survey of these results since they are interesting in their own right and shed some light on the probabilistic behavior of pattern matching based data compression

markov types and minimax redundancy for markov sources redundancy of universal codes for a class of sources determines by how much the actual code length exceeds the optimal code length in the minimax scenario one designs the best code for the worst source within the class such minimax redundancy comes in two flavors average minimax or worst case minimax we study the worst case minimax redundancy of universal block codes for markovian sources of any order we prove that the maximal minimax redundancy for markov sources of order r is asymptotically equal to 1 2m sup r m 1 log sub 2 n log sub 2 a sub m sup r lnlnm sup 1 m 1 lnm o 1 where n is the length of a source sequence m is the size of the alphabet and a sub m sup r is an explicit constant e g we find that for a binary alphabet m 2 and markov of order r 1 the constant a sub 2 sup 1 16 spl middot g spl ap 14 655449504 where g is the catalan number unlike previous attempts we view the redundancy problem as an asymptotic evaluation of certain sums over a set of matrices representing markov types the enumeration of markov types is accomplished by reducing it to counting eulerian paths in a multigraph in particular we propose exact and asymptotic formulas for the number of strings of a given markov type all of these findings are obtained by analytic and combinatorial tools of analysis of algorithms

asymptotic average redundancy of huffman and other block codes we study asymptotically the redundancy of huffman and other codes it has been known from the inception of the huffman 1952 code that in the worst case its redundancy defined as the excess of the code length over the optimal ideal code length is not more than one however to the best of our knowledge no precise asymptotic results have been reported in literature thus far we consider here a memoryless binary source generating a sequence of length n distributed as binomial n p with p being the probability of emitting 0 based on the results of stubley 1994 we prove that for p 1 2 1 m 1 2 sup 1 m 2 sup m o spl rho sup n spl alpha n m rational where spl alpha log sub 2 1 p p and spl beta log sub 2 1 p spl rho x x is the fractional part of x the appearance of the fractal like function explains the erratic behavior of the huffman redundancy and its resistance to succumb to a precise analysis as a side result we prove that the average redundancy of the shannon block code is as n spl rarr spl infin r sub n sup s 1 2 o 1 spl alpha irrational 1 2 1 m 1 2 o spl rho sup n spl alpha n m rational where spl rho 1 finally we derive the redundancy of the golomb 1966 code for the geometric distribution which can be viewed as a special case of the huffman and shannon codes golomb s code redundancy has only oscillating behavior i e there is not convergent mode these findings are obtained by analytic methods such as theory of distribution of sequences modulo 1 and fourier series

on the performance of non orthogonal multiple access systems with partial channel information in this paper a downlink single cell non orthogonal multiple access noma network with uniformly deployed users is considered and an analytical framework to evaluate its performance is developed particularly the performance of noma is studied by assuming two types of partial channel state information csi for the first one which is based on imperfect csi we present a simple closed form approximation for the outage probability and the average sum rate as well as their high signal to noise ratio snr expressions for the second type of csi which is based on second order statistics sos we derive a closed form expression for the outage probability and an approximate expression for the average sum rate for the special case two users for the addressed scenario with the two types of partial csi the results demonstrate that noma can achieve superior performance compared to the traditional orthogonal multiple access oma moreover sos based noma always achieves better performance than that with imperfect csi while it can achieve similar performance to the noma with perfect csi at the low snr region the provided numerical results confirm that the derived expressions for the outage probability and the average sum rate match well with the monte carlo simulations

intermittency in passive scalar decay we consider the decay of a passive scalar in a rough turbulent flow we show that at large times the scalar intensity is statistically equivalent to the product of two independent random variables the first factor is a gaussian variable related to the typical fluctuation at initial conditions the second factor is essentially the probability that in a given realization of the flow tracer particles separate at a very slow rate therefore the large time scalar distribution is always broader than gaussian for any initial condition and large scalar excursions are associated to small relative dispersion events

anomalous scaling in anisotropic turbulence abstract we present a short review of the work conducted by our group on the subject of anomalous scaling in anisotropic turbulence the basic idea that unifies all the applications discussed here is that the equations of motion for correlation functions are always linear and invariant to rotations and therefore the solutions foliate into sectors of the symmetry group of all rotations so 3 we have considered models of passive scalar and passive vector advections by a rapidly changing turbulent velocity field kraichnan type models for which we find a discrete spectrum of universal anomalous exponents with a different exponent characterizing the scaling behavior in every sector generically the correlation functions and structure functions appear as sums over all these contributions with nonuniversal amplitudes which are determined by the anisotropic boundary conditions in addition we considered navier stokes turbulence by analyzing simulations and experiments and reached some interesting conclusions regarding the scaling exponents in the anisotropic sectors the theory presented here clarifies questions like the restoration of local isotropy upon decreasing scales we explain when the local isotropy is fully restored and when the lingering effects of the anisotropic forcing appear for arbitrarily small scales

connection between nonlinear energy optimization and instantons how systems transit between different stable states under external perturbation is an important practical issue we discuss here how a recently developed energy optimization method for identifying the minimal disturbance necessary to reach the basin boundary of a stable state is connected to the instanton trajectory from large deviation theory of noisy systems in the context of the one dimensional swift hohenberg equation which has multiple stable equilibria we first show how the energy optimization method can be straightforwardly used to identify minimal disturbances minimal seeds for transition to specific attractors from the ground state then after generalizing the technique to consider multiple equally spaced in time perturbations it is shown that the instanton trajectory is indeed the solution of the energy optimization method in the limit of infinitely many perturbations provided a specific norm is used to measure the set of discrete perturbations importantly we find that the key features of the instanton can be captured by a low number of discrete perturbations typically one perturbation per basin of attraction crossed this suggests a promising new diagnostic for systems for which it may be impractical to calculate the instanton

evolution of a scalar gradient s probability density function in a random flow the evolution of the probability density function pdf of the passive scalar gradient is studied in the limit of large peclet and prandtl numbers in d dimensions without diffusion the closed fokker planck equation can be derived and solved analytically leading to a number of conclusions in particular it allows the description of the restoration of the rotational symmetry and enables one to distinguish different regimes of evolution and different intervals of pdf behavior

spontaneous stochasticity and anomalous dissipation for burgers equation we develop a lagrangian approach to conservation law anomalies in weak solutions of inviscid burgers equation motivated by previous work on the kraichnan model of turbulent scalar advection we show that the entropy solutions of burgers possess markov stochastic processes of generalized lagrangian trajectories backward in time for which the burgers velocity is a backward martingale this property is shown to guarantee dissipativity of conservation law anomalies for general convex functions of the velocity the backward stochastic burgers flows with these properties are not unique however we construct infinitely many such stochastic flows both by a geometric construction and by the zero noise limit of the constantin iyer stochastic representation of viscous burgers solutions the latter proof yields the spontaneous stochasticity of lagrangian trajectories backward in time for burgers at unit prandtl number it is conjectured that existence of a backward stochastic flow with the velocity as martingale is an admissibility condition which selects the unique entropy solution for burgers we also study linear transport of passive densities and scalars by inviscid burgers flows we show that shock solutions of burgers exhibit spontaneous stochasticity backward in time for all finite prandtl numbers implying conservation law anomalies for linear transport we discuss the relation of our results for burgers with incompressible navier stokes turbulence especially lagrangian admissibility conditions for euler solutions and the relation between turbulent cascade directions and time asymmetry of lagrangian stochasticity

study of the hopf functional equation for turbulence duhamel principle and dynamical scaling we consider a formulation for the hopf functional differential equation which governs statistical solutions of the navier stokes equations by introducing an exponential operator with a functional derivative we recast the hopf equation as an integro differential functional equation by the duhamel principle on this basis we introduce a successive approximation to the hopf equation as an illustration we take the burgers equation and carry out the approximations to the leading order scale invariance of the statistical navier stokes equations in d dimensions is formulated and contrasted with that of the deterministic navier stokes equations for the statistical navier stokes equations critical scale invariance is achieved for the characteristic functional of the d th derivative of the vector potential in d dimensions the deterministic equations corresponding to this choice of the dependent variable acquire the linear fokker planck operator under dynamic scaling in three dimensions it is the vorticity gradient that behaves like a fundamental solution more precisely source type solution of deterministic navier stokes equations in the long time limit physical applications of these ideas include study of a self similar decaying profile of fluid flows moreover we reveal typical physical properties in the late stage evolution by combining statistical scale invariance and the source type solution this yields an asymptotic form of the hopf functional in the long time limit improving the well known hopf titt solution in particular we present analyses for the burgers equations to illustrate the main ideas and indicate a similar analysis for the navier stokes equations

average redundancy of the shannon code for markov sources it is known that for memoryless sources the average and maximal redundancy of fixed to variable length codes such as the shannon and huffman codes exhibit two modes of behavior for long blocks it either converges to a limit or it has an oscillatory pattern depending on the irrationality or rationality respectively of certain parameters that depend on the source in this paper we extend these findings concerning the shannon code to the case of a markov source we provide a precise characterization of the convergent versus oscillatory behavior of the shannon code redundancy for a class of irreducible periodic and aperiodic markov sources these findings are obtained by analytic methods such as fourier fejer series analysis and spectral analysis of matrices

instantons and intermittency we describe the method for finding the non gaussian tails of the probability distribution function pdf for solutions of a stochastic differential equation such as the convection equation for a passive scalar the random driven navier stokes equation etc the existence of such tails is generally regarded as a manifestation of the intermittency phenomenon our formalism is based on the wkb approximation in the functional integral for the conditional probability of large fluctuation we argue that the main contribution to the functional integral is given by a coupled field force configuration char22 the instanton as an example we examine the correlation functions of the passive scalar u advected by a large scale velocity field ensuremath delta correlated in time we find the instanton determining the tails of the generating functional and show that it is different from the instanton that determines the probability distribution function of high powers of u we discuss the simplest instantons for the navier stokes equation textcopyright 1996 the american physical society

rigorous estimates of the tails of the probability distribution function for the random linear shear model in previous work majda and mclaughlin and majda computed explicit expressions for the 2nth moments of a passive scalar advected by a linear shear flow in the form of an integral over r n in this paper we first compute the asymptotics of these moments for large moment number we are able to use this information about the large n behavior of the moments along with some basic facts about entire functions of finite order to compute the asymptotics of the tails of the probability distribution function we find that the probability distribution has gaussian tails when the energy is concentrated in the largest scales as the initial energy is moved to smaller and smaller scales we find that the tails of the distribution grow longer and the distribution moves smoothly from gaussian through exponential and stretched exponential we also show that the derivatives of the scalar are increasingly intermittent in agreement with experimental observations and relate the exponents of the scalar derivative to the exponents of the scalar

diffusing passive tracers in random incompressible flows statistical topography aspects the paper studies statistical characteristics of the passive tracer concentrations and of its spatial gradient in random incompressible velocity fields from the viewpoint of statistical topography the statistics of interest include mean values probability distributions as well as various functionals characterizing topographic features of tracers the functional approach is used we consider the influence of the mean flow the linear shear flow and the molecular diffusion coefficient on the statistics of the tracer most of our analysis is carried out in the framework of the delta correlated in time approximation and conditions for its applicability are established but we also consider the diffusion approximation scheme for finite correlation radius the latter is applied to a diffusing passive tracer that undergoes sedimentation in a random velocity field

large scale probability density function for scalar field advected by high reynolds number turbulent flow a closed equation for the one point probability density function pdf for a scalar field advected by the three dimensional random velocity field with arbitrary many spatial temporal scales and infrared divergence in the limit of high reynolds number has been derived using the functional derivative technique and renormalization theory together with the assumption that the turbulent velocity is a homogeneous isotropic gaussian random field it has been shown that when the spectral parameters of a random velocity field slightly deviate from their kolmogorov obukhov values the equation for the pdf in the long time large distance limit can be derived exactly and well described by a conventional diffusion theory while the lagrangian scaling function describing the large scale particle displacements in turbulent flow is essentially superdiffusive the scaling procedure in the limit of high reynolds number allows us to completely overcome the well known closure problem associated with diffusion term

mathematical models with exact renormalization for turbulent transport the advection diffusion of a passive scalar by incompressible velocity fields which admit a statistical description and involve a continuous range of excited spatial and or temporal scales is very important in applications ranging from fully developed turbulence to the diffusion of tracers in heterogeneous porous media a variety of renormalization theories which typically utilize partial resummation of divergent perturbation series according to various recipes have been applied to this problem in various contexts in this paper a simple model problem for the advection diffusion of a passive scalar is introduced and the complete renormalization theory is developed with full mathematical rigor explicit formulas for the anomalous time scaling in various regimes as well as the green s function for the large scale long time ensemble average are developed here formulas for the renormalized higher order statistics are also developed the simple form of the model problem is deceptive the renormalization theory for this problem exhibits a remarkable range of different renormalization phenomena as parameters in the velocity statistics are varied these phenomena include the existence of several distinct anomalous scaling regimes as the spectral parameter tilde varepsilon is varied as well as explicit regimes in tilde varepsilon where the effective equation for the ensemble average is not a simple diffusion equation but instead involves an explicit random nonlocal eddy diffusivity we use fourier analysis and the feynman kac formula as main tools in the explicit exact renormalization theory developed here

scalar flux in a uniform mean scalar gradient in homogeneous isotropic steady turbulence abstract statistics of a passive scalar flux in a uniform mean scalar gradient convected by homogeneous isotropic steady turbulence are numerically studied by using very high resolution direct numerical simulation it is found that the nusselt number increases in proportion to the peclet number and that the one point probability density function of the scalar flux is negatively skewed and exponential and is insensitive to the variation of the peclet number the scalar field is studied by visualization and the ramp cliff structure and the mesa canyon structure are observed along the directions parallel and perpendicular to the mean scalar gradient respectively the probability density function of the scalar flux is theoretically computed and found to be in good agreement with the numerical results a lagrangian statistical theory for the scalar flux is developed which predicts that the scalar transfer flux is given by the time integral of the lagrangian velocity autocorrelation and increases in proportion to the peclet number which is consistent with the results of the direct numerical simulation a physical explanation of the asymmetry of the scalar flux pdf is explored

evolution of the probability measure for the majda model new invariant measures and breathing pdfs in 1993 majda proposed a simple random shear model from which scalar inter mittency was rigorously predicted for the invariant probability measure of passive tracers in this work we present an integral formulation for the tracer measure which leads to a new comprehensive study on its temporal evolution based on monte carlo simulation and direct numerical integration an interesting non monotonic breathing phenomenon is discov ered from these results and carefully defined with a solid example for special initial data to predict such phenomenon the signature of this phenomenon may persist at long time char acterized by the approach of the pdf core to its infinite time invariant value we find that this approach may be strongly dependent on the non dimensional peclet number of which the invariant measure itself is independent further the breathing pdf is recovered as a new invariant measure in a distinguished time scale in the diffusionless limit rigorous as ymptotic analysis is also performed to identify the gaussian core of the invariant measures and the critical rate at which the heavy stretched exponential regime propagates towards the tail as a function of time is calculated

intermittency and anomalous scaling of passive scalars in any space dimension we establish exact inequalities for the structure function scaling exponents of a passively advected scalar in both the inertial convective and viscous convective ranges these inequalities involve the scaling exponents of the velocity structure functions and in a refined form an intermittency exponent of the convective range scalar flux they are valid for three dimensional navier stokes turbulence and satisfied within errors by present experimental data the inequalities also hold for any synthetic turbulent velocity statistics with a finite correlation in time we show that for time correlation exponents of the velocity smaller than the local turnover exponent the scalar spectral exponent is strictly less than that in kraichnan s phys rev lett 72 1016 1994 phys fluids 11 945 1968 j fluid mech 64 737 1974 62 305 1974 soluble rapid change model with velocity ensuremath delta correlated in time our results include as a special case an exponent inequality derived previously by constantin and procaccia nonlinearity 7 1045 1994 but with a more direct proof the inequalities in their simplest form follow from a kolmogorov type relation for the turbulent passive scalar valid in each space dimension d our improved inequalities are based upon a rigorous version of the refined similarity hypothesis for passive scalars these are compared with the relations implied by fusion rules hypothesized for scalar gradients textcopyright 1996 the american physical society

universal long time properties of lagrangian statistics in the batchelor regime and their application to the passive scalar problem we consider the transport of dynamically passive quantities in the batchelor regime of a smooth in space velocity field for the case of arbitrary temporal correlations of the velocity we formulate the statistics of relevant characteristics of lagrangian motion this allows us to generalize many results obtained previously for strain ensuremath delta correlated in time thus answering a question about the universality of these results

complexity of strings in the class of markov sources shannon s self information of a string is generalized to its complexity relative to the class of finite state machine fsm defined sources unlike an earlier generalization the new one is valid for both short and long strings the definition is justified in part by a theorem stating that asymptotically the mean complexity provides a tight lower bound for the mean length of all so called regular codes this also generalizes shannon s noiseless coding theorem for a large subclass of fsm sources a simple algorithm is described for computing the complexity

quantization noise spectra several results describing the behavior of quantization noise in a unified and simplified manner are discussed exact formulas for quantizer noise spectra are developed they are applied to a variety of systems and inputs including scalar quantization pcm dithered pcm sigma delta modulation dithered sigma delta modulation two stage sigma delta modulation and second order sigma delta modulation

outage probability in arbitrarily shaped finite wireless networks this paper analyzes the outage performance in finite wireless networks unlike most prior works which either assumed a specific network shape or considered a special location of the reference receiver we propose two general frameworks for analytically computing the outage probability at any arbitrary location of an arbitrarily shaped finite wireless network i a moment generating function based framework which is based on the numerical inversion of the laplace transform of a cumulative distribution and ii a reference link power gain based framework which exploits the distribution of the fading power gain between the reference transmitter and receiver the outage probability is spatially averaged over both the fading distribution and the possible locations of the interferers the boundary effects are accurately accounted for using the probability distribution function of the distance of a random node from the reference receiver for the case of the node locations modeled by a binomial point process and nakagami m fading channel we demonstrate the use of the proposed frameworks to evaluate the outage probability at any location inside either a disk or polygon region the analysis illustrates the location dependent performance in finite wireless networks and highlights the importance of accurately modeling the boundary effects