shift in the velocity of a front due to a cutoff we consider the effect of a small cutoff ensuremath varepsilon on the velocity of a traveling wave in one dimension simulations done over more than ten orders of magnitude as well as a simple theoretical argument indicate that the effect of the cutoff ensuremath varepsilon is to select a single velocity that converges when ensuremath varepsilon ensuremath rightarrow 0 to the one predicted by the marginal stability argument for small ensuremath varepsilon the shift in velocity has the form k mathrm ln ensuremath varepsilon ensuremath 2 and our prediction for the constant k agrees very well with the results of our simulations a very similar logarithmic shift appears in more complicated situations in particular in finite size effects of some microscopic stochastic systems our theoretical approach can also be extended to give a simple way of deriving the shift in position due to initial conditions in the fisher kolmogorov or similar equations

duality in interacting particle systems and boson representation in the context of markov processes we show a new scheme to derive dual processes and a duality function based on a boson representation this scheme is applicable to a case in which a generator is expressed by boson creation and annihilation operators for some stochastic processes duality relations have been known which connect continuous time markov processes with discrete state space and those with continuous state space we clarify that using a generating function approach and the doi peliti method a birth death process or discrete random walk model is naturally connected to a differential equation with continuous variables which would be interpreted as a dual markov process the key point in the derivation is to use bosonic coherent states as a bra state instead of a conventional projection state as examples we apply the scheme to a simple birth coagulation process and a brownian momentum process the generator of the brownian momentum process is written by elements of the su 1 1 algebra and using a boson realization of su 1 1 we show that the same scheme is available

phenomenological theory giving the full statistics of the position of fluctuating pulled fronts we propose a phenomenological description for the effect of a weak noise on the position of a front described by the fisher kolmogorov petrovsky piscounov equation or any other traveling wave equation in the same class our scenario is based on four hypotheses on the relevant mechanism for the diffusion of the front our parameter free analytical predictions for the velocity of the front its diffusion constant and higher cumulants of its position agree with numerical simulations

speed and fluctuations of n particle branching brownian motion with spatial selection we consider branching brownian motion on the real line with the following selection mechanism every time the number of particles exceeds a large given number n only the n right most particles are kept and the others killed after rescaling time by log 3n we show that the properly recentred position of the lceil alpha n rceil th particle from the right alpha in 0 1 converges in law to an explicitly given spectrally positive levy process this behaviour has been predicted to hold for a large class of models falling into the universality class of the fkpp equation with weak multiplicative noise brunet et al in phys rev e 73 5 056126 2006 and is proven here for the first time for such a model

universal tree structures in directed polymers and models of evolving populations by measuring or calculating coalescence times for several models of coalescence or evolution with and without selection we show that the ratios of these coalescence times become universal in the large size limit and we identify a few universality classes

extended duality relations between birth death processes and partial differential equations duality relations between continuous state and discrete state stochastic processes with continuous time have already been studied and used in various research fields we propose extended duality relations which enable us to derive discrete state stochastic processes from arbitrary diffusion type partial differential equations the derivation is based on the doi peliti formalism and it will be clarified that additional states for the discrete state stochastic processes must be considered in some cases

probabilistic motion estimation based on temporal coherence we develop a theory for the temporal integration of visual motion motivated by psychophysical experiments the theory proposes that input data are temporally grouped and used to predict and estimate the motion flows in the image sequence this temporal grouping can be considered a generalization of the data association techniques that engineers use to study motion sequences our temporal grouping theory is expressed in terms of the bayesian generalization of standard kalman filtering to implement the theory we derive a parallel network that shares some properties of cortical networks computer simulations of this network demonstrate that our theory qualitatively accounts for psychophysical experiments on motion occlusion and motion outliers in deriving our theory we assumed spatial factorizability of the probability distributions and made the approximation of updating the marginal distributions of velocity at each point this allowed us to perform local computations and simplified our implementation we argue that these approximations are suitable for the stimuli we are considering for which spatial coherence effects are negligible

effect of selection on ancestry an exactly soluble case and its phenomenological generalization we consider a family of models describing the evolution under selection of a population whose dynamics can be related to the propagation of noisy traveling waves for one particular model that we shall call the exponential model the properties of the traveling wave front can be calculated exactly as well as the statistics of the genealogy of the population one striking result is that for this particular model the genealogical trees have the same statistics as the trees of replicas in the parisi mean field theory of spin glasses we also find that in the exponential model the coalescence times along these trees grow like the logarithm of the population size a phenomenological picture of the propagation of wave fronts that we introduced in a previous work as well as our numerical data suggest that these statistics remain valid for a larger class of models while the coalescence times grow like the cube of the logarithm of the population size

c ondensation conditional density propagation forvisual tracking the problem of tracking curves in dense visual clutter is challenging kalman filtering is inadequate because it is based on gaussian densities which being unimo dal cannot represent simultaneous alternative hypotheses the condensation algorithm uses factored sampling previously applied to the interpretation of static images in which the probability distribution of possible interpretations is represented by a randomly generated set condensation uses learned dynamical models together with visual observations to propagate the random set over time the result is highly robust tracking of agile motion notwithstanding the use of stochastic methods the algorithm runs in near real time

adaptive walks and extreme value theory we study biological evolution in a high dimensional genotype space in the regime of rare mutations and strong selection the population performs an uphill walk which terminates at local fitness maxima assigning fitness randomly to genotypes we show that the mean walk length is logarithmic in the number of initially available beneficial mutations with a prefactor determined by the tail of the fitness distribution this result is derived analytically in a simplified setting where the mutational neighborhood is fixed during the adaptive process and confirmed by numerical simulations

universality classes of interaction structures for nk fitness landscapes kauffman s nk model is a paradigmatic example of a class of stochastic models of genotypic fitness landscapes that aim to capture generic features of epistatic interactions in multilocus systems genotypes are represented as sequences of l binary loci the fitness assigned to a genotype is a sum of contributions each of which is a random function defined on a subset of k le l loci these subsets or neighborhoods determine the genetic interactions of the model whereas earlier work on the nk model suggested that most of its properties are robust with regard to the choice of neighborhoods recent work has revealed an important and sometimes counter intuitive influence of the interaction structure on the properties of nk fitness landscapes here we review these developments and present new results concerning the number of local fitness maxima and the statistics of selectively accessible that is fitness monotonic mutational pathways in particular we develop a unified framework for computing the exponential growth rate of the expected number of local fitness maxima as a function of l and identify two different universality classes of interaction structures that display different asymptotics of this quantity for large k moreover we show that the probability that the fitness landscape can be traversed along an accessible path decreases exponentially in l for a large class of interaction structures that we characterize as locally bounded finally we discuss the impact of the nk interaction structures on the dynamics of evolution using adaptive walk models

direct nonrecursive relations between cepstrum and predictor coefficients direct i e nonrecursive relations are derived for the cepstrum in terms of the predictor coefficients and vice versa connections with algebraic roots symmetric functions statistical moments and cumulants are pointed out some implications for pitch detection are also discussed

on travelling waves for the stochastic fisher kolmogorov petrovsky piscunov equation this paper is concerned with properties of the wave speed for the stochastically perturbed fisher kolmogorov petrovsky piscunov fkpp equation it was shown in the classical 1937 paper by kolmogorov petrovsky and piscunov that the large time behavior of the solution to the fkpp equation with heaviside initial data is a travelling wave in a seminal 1995 paper mueller and sowers proved that this also holds for a stochastically perturbed fkpp equation the wave speed depends on the strength of the noise in this paper bounds on the asymptotic behavior of the wave speed c as 0 and are obtained

mathematical properties of a class of integro differential models from population genetics we propose a mathematical analysis of an integro differential model arising in population genetics the model describes the dynamics of fitness distribution in an asexual population under the effect of mutation and selection these two processes are represented by two nonlocal terms first we prove the existence and uniqueness of the solution and we derive asymptotic estimates of the distribution as the fitness tends to pm infty based on these asymptotic estimates we show that the cumulant generating function of the distribution is well defined and satisfies a linear nonlocal transport equation that we solve explicitly this explicit formula allows us to characterize the dependence of the long time behavior of the distribution with respect to the mutation kernel on the one hand if the kernel contains some beneficial mutations the distribution diverges which is reminiscent of the results of alfaro and carles siam j appl math 74 2014 pp 1919 1934 who analyzed a mutator replicator equat

calculating the complex cepstrum without phase unwrapping or integration the complex cepstrum as developed by oppenheim 4 and oppenheim schafer and stockham 5 originated from the pioneering paper of bogert healy and tukey 3 it is a nonlinear transformation from one signal space into another which converts two superposed signals into their sum the cepstrum is usually computed by inverse fourier transformation of the frequency domain complex logarithm of the original signal the phase term or imaginary part of this complex logarithm presents a computational difficulty which is sometimes approached by integration of the derivative of the phase spectrum an excellent approach to phase computation is the adaptive integration methodology as proposed by tribolet 10 unfortunately this technique can be computationally intensive this note describes a simple technique for efficient computation of the complex cepstrum which requires no phase unwrapping or integration the procedure is given in both one and two dimensions

evolution on a smooth landscape we study in detail a recently proposed simple discrete model for evolution on smooth landscapes an asymptotic solution of this model for long times is constructed we find that the dynamics of the population is governed by correlation functions that although being formally down by powers ofn the population size nonetheless control the evolution process after a very short transient the long time behavior can be found analytically since only one of these higher order correlators the two point function is relevant we compare and contrast the exact findings derived herein with a previously proposed phenomenological treatment employing mean field theory supplemented with a cutoff at small population density finally we relate our results to the recently studied case of mutation on a totally flat landscape

interacting particles the stochastic fisher kolmogorov petrovsky piscounov equation and duality abstract the stochastic fisher kolmogorov petrovsky piscunov equation is t u x t d xx u u 1 u e u 1 u x t for 0 u 1 where x t is a gaussian white noise process in space and time here d and e are parameters and the equation is interpreted as the continuum limit of a spatially discretized set of ito equations solutions of this stochastic partial differential equation have an exact connection to the a a a reaction diffusion system at appropriate values of the rate coefficients and particles diffusion constant this relationship is called duality by the probabilists it is not via some hydrodynamic description of the interacting particle system in this paper we present a complete derivation of the duality relationship and use it to deduce some properties of solutions to the stochastic fisher kolmogorov petrovsky piscunov equation

the diffusion coefficient of propagating fronts with multiplicative noise recent studies have shown that in the presence of noise both fronts propagating into a metastable state and so called pushed fronts propagating into an unstable state exhibit diffusive wandering about the average position in this paper we derive an expression for the effective diffusion coefficient of such fronts which was motivated before on the basis of a multiple scale ansatz our systematic derivation is based on the decomposition of the fluctuating front into a suitably positioned average profile plus fluctuating eigenmodes of the stability operator while the fluctuations of the front position in this particular decomposition are a wiener process on all time scales the fluctuations about the time averaged front profile relax exponentially

anderson transition on the cayley tree as a traveling wave critical point for various probability distributions for anderson localization on the cayley tree we study the statistics of various observables as a function of the disorder strength w and the number n of generations we first consider the landauer transmission tn in the localized phase its logarithm follows the traveling wave form where i the disorder averaged value moves linearly and the localization length diverges as with loc 1 and ii the variable t is a fixed random variable with a power law tail p t 1 t 1 w for large t with 0 w 1 2 so that all integer moments of tn are governed by rare events in the delocalized phase the transmission tn remains a finite random variable as n and we measure near criticality the essential singularity with t 0 25 we then consider the statistical properties of normalized eigenstates x x 2 1 in particular the entropy s x x 2ln x 2 and the inverse participation ratios ipr iq x x 2q in the localized phase the typical entropy diverges as with s 1 5 whereas it grows linearly as styp n n in the delocalized phase finally for the ipr we explain how closely related variables propagate as traveling waves in the delocalized phase in conclusion both the localized phase and the delocalized phase are characterized by the traveling wave propagation of some probability distributions and the anderson localization delocalization transition then corresponds to a traveling non traveling critical point moreover our results point toward the existence of several length scales that diverge with different exponents at criticality

extreme value statistics and traveling fronts various applications abstract an intriguing connection between extreme value statistics and traveling fronts has been found recently in a number of diverse problems in this short review we outline a few such problems and consider their various applications

adaptive evolution on a continuous lattice model in the current work we investigate the evolutionary dynamics of a spatially structured population model defined on a continuous lattice in the model individuals disperse at a constant rate v and competition is local and delimited by the competition radius r due to dispersal the neighborhood size number of individuals competing for reproduction fluctuates over time here we address how these new variables affect the adaptive process while the fixation probabilities of beneficial mutations are roughly the same as in a panmitic population for small fitness effects s a dependence on v and r becomes more evident for large s these quantities also strongly influence fixation times but their dependencies on s are well approximated by s ensuremath 1 2 which means that the speed of the genetic wave front is proportional to sqrt s most important is the observation that the model exhibits a dual behavior displaying a power law growth for the fixation rate and speed of adaptation with the beneficial mutation rate as observed in other spatially structured population models while simultaneously showing a nonsaturating behavior for the speed of adaptation with the population size n as in homogeneous populations

effect of noise on front propagation in reaction diffusion equations of kpp type we consider reaction diffusion equations of kpp type in one spatial dimension perturbed by a fisher wright white noise under the assumption of uniqueness in distribution examples include the randomly perturbed fisher kpp equations partial tu partial x 2u u 1 u epsilon sqrt u 1 u dot w and partial tu partial x 2u u 1 u epsilon sqrt u dot w where dot w dot w t x is a space time white noise we prove the brunet derrida conjecture that the speed of traveling fronts for small e is 2 pi 2 log epsilon 2 2 o log log epsilon log epsilon 3

analysis of evolution through competitive selection recent studies of in vitro evolution of dna via protein binding indicate that the evolution behavior is qualitatively different in different parameter regimes i here present a general theory that is valid for a wide range of parameters and which reproduces and extends previous results specifically the mean field theory of a general translation invariant model can be reduced to the basic diffusion equation with a dynamic boundary condition the simple analytical form yields both quantitatively accurate predictions and valuable insight into the principles involved

asymptotic scaling of the diffusion coefficient of fluctuating pulled fronts we present a heuristic theoretical derivation for the scaling of the diffusion coefficient d f for fluctuating pulled fronts in agreement with earlier numerical simulations we find that as n infinity d f approaches zero as 1 ln 3 n where n is the average number of particles per correlation volume in the stable phase of the front this behavior of d f stems from the shape fluctuations at the very tip of the front and is independent of the microscopic model

field theory of propagating reaction diffusion fronts the problem of velocity selection of reaction diffusion fronts has been widely investigated while the mean field limit results are well known theoretically there is a lack of analytic progress in those cases in which fluctuations are to be taken into account here we construct an analytic theory connecting the first principles of the reaction diffusion process to an effective equation of motion via field theoretic arguments and we arrive at results already confirmed by numerical simulations

speed of pulled fronts with a cutoff we study the effect of a small cutoff epsilon on the velocity of a pulled front in one dimension by means of a variational principle we obtain a lower bound on the speed dependent on the cutoff for which the two leading order terms correspond to the brunet derrida expression to do so we cast a known variational principle for the speed of propagation of fronts in different variables which makes it more suitable for applications

large deviations for the branching brownian motion in presence of selection or coalescence the large deviation function has been known for a long time in the literature for the displacement of the rightmost particle in a branching random walk brw or in a branching brownian motion bbm more recently a number of generalizations of the bbm and of the brw have been considered where selection or coalescence mechanisms tend to limit the exponential growth of the number of particles here we try to estimate the large deviation function of the position of the rightmost particle for several such generalizations the l bbm the n bbm and the coalescing branching random walk cbrw which is closely related to the noisy fkpp equation our approach allows us to obtain only upper bounds on these large deviation functions one noticeable feature of our results is their non analytic dependence on the parameters such as the coalescence rate in the cbrw

motion based prediction is sufficient to solve the aperture problem in low level sensory systems it is still unclear how the noisy information collected locally by neurons may give rise to a coherent global percept this is well demonstrated for the detection of motion in the aperture problem as luminance of an elongated line is symmetrical along its axis tangential velocity is ambiguous when measured locally here we develop the hypothesis that motion based predictive coding is sufficient to infer global motion our implementation is based on a context dependent diffusion of a probabilistic representation of motion we observe in simulations a progressive solution to the aperture problem similar to physio logy and behavior we demonstrate that this solution is the result of two underlying mechanisms first we demonstrate the formation of a tracking behavior favoring temporally coherent features independent of their texture second we observe that incoherent features are explained away while coherent information diffuses progressively to the global scale most previous models included ad hoc mechanisms such as end stopped cells or a selection layer to track specific luminance based features as necessary conditions to solve the aperture problem here we have proved that motion based predictive coding as it is implemented in this functional model is sufficient to solve the aperture problem this solution may give insights into the role of prediction underlying a large class of sensory computations

time domain cepstral transformations the realization of homomorphic systems for convolution is addressed motivated by the limitations of the fourier transform based method commonly used in homomorphic complex cepstral filtering the time domain cepstral transformation tdct method which is entirely based on time domain calculations thus avoiding or minimizing the problems associated with the ft method is introduced explicit transformations of an ordinary mixed phase time sequence belonging to convolution space into its complex cepstrum time sequence belonging to additive cepstrum vector space and vice versa are derived the method does not require unwrapped phase calculations and no specific windows are used to precondition the signal in order to produce a more accurate representation of the complex cepstrum it trades reduced computational efficiency for improved performance examples comparing fourier based and td cepstrum transformations are presented

interfacial velocity corrections due to multiplicative noise the problem of velocity selection for reaction fronts has been intensively investigated leading to the successful marginal stability ms approach for propagation into an unstable state because the front velocity is controlled by the leading edge which perforce has low density it is interesting to study the role that finite particle number fluctuations have on this picture here we use the well known mapping of discrete markov processes to stochastic differential equations and focus on the front velocity in the simple a a ensuremath leftrightarrows a system our results are consistent with a recent heuristic proposal that v mathrm ms ensuremath v ensuremath sim 1 mathrm ln 2 n

random travelling waves for the kpp equation with noise consider the stochastic partial differential equation formula where t x is two parameter while noise assume that u0 is a continuous function taking values in 0 1 such that for some constant a 0 we have c1 u0 x 1 for x a let the wavefront b t sup x r u t x 0 we show that for small enough and with probability 1 limt b t t exists and lies in 0 this limit depends only on the law of v t x u t b t x tends toward a stationary limit as t we also analyze the length of the region a t b t which is the smallest closed interval containing the points x at which 0 u t x 1 we show that the length of this region tends toward a stationary distribution thus the wavefront does not degenerate

the noisy edge of traveling waves traveling waves are ubiquitous in nature and control the speed of many important dynamical processes including chemical reactions epidemic outbreaks and biological evolution despite their fundamental role in complex systems traveling waves remain elusive because they are often dominated by rare fluctuations in the wave tip which have defied any rigorous analysis so far here we show that by adjusting nonlinear model details noisy traveling waves can be solved exactly the moment equations of these tuned models are closed and have a simple analytical structure resembling the deterministic approximation supplemented by a nonlocal cutoff term the peculiar form of the cutoff shapes the noisy edge of traveling waves and is critical for the correct prediction of the wave speed and its fluctuations our approach is illustrated and benchmarked using the example of fitness waves arising in simple models of microbial evolution which are highly sensitive to number fluctuations we demonstrate explicitly how these models can be tuned to account for finite population sizes and determine how quickly populations adapt as a function of population size and mutation rates more generally our method is shown to apply to a broad class of models in which number fluctuations are generated by branching processes because of this versatility the method of model tuning may serve as a promising route toward unraveling universal properties of complex discrete particle systems

the speed of evolution in large asexual populations we consider an asexual biological population of constant size n evolving in discrete time under the influence of selection and mutation beneficial mutations appear at rate u and their selective effects s are drawn from a distribution g s after introducing the required models and concepts of mathematical population genetics we review different approaches to computing the speed of logarithmic fitness increase as a function of n u and g s we present an exact solution of the infinite population size limit and provide an estimate of the population size beyond which it is valid we then discuss approximate approaches to the finite population problem distinguishing between the case of a single selection coefficient g s s s b and a continuous distribution of selection coefficients analytic estimates for the speed are compared to numerical simulations up to population sizes of order 10300

an exactly solvable travelling wave equation in the fisher kpp class for a simple one dimensional lattice version of a travelling wave equation we obtain an exact relation between the initial condition and the position of the front at any later time this exact relation takes the form of an inverse problem given the times t n at which the travelling wave reaches the positions n one can deduce the initial profile we show by means of complex analysis that a number of known properties of travelling wave equations in the fisher kpp class can be recovered in particular bramson s shifts of the positions we also recover and generalize ebert van saarloos corrections depending on the initial condition

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

mosaic length and finite interaction range effects in a one dimensional random energy model in this paper we study finite interaction range corrections to the mosaic picture of the glass transition as it emerges from the study of the kac limit of large interaction range for disordered models to this aim we consider point to set correlation functions or overlaps in a one dimensional random energy model as a function of the range of interaction in the kac limit the mosaic length defines a sharp first order transition separating a high overlap phase from a low overlap one correspondingly we find that overlap curves as a function of the window size and different finite interaction ranges cross roughly at the mosaic length nonetheless we find a very slow convergence to the kac limit and discuss why this could be a problem for measuring the mosaic length in realistic models

exactly soluble noisy traveling wave equation appearing in the problem of directed polymers in a random medium we calculate exactly the velocity and diffusion constant of a microscopic stochastic model of n evolving particles which can be described by a noisy traveling wave equation with a noise of order n 1 2 our model can be viewed as the infinite range limit of a directed polymer in random medium with n sites in the transverse direction despite some peculiarities of the traveling wave equations in the absence of noise our exact solution allows us to test the validity of a simple cutoff approximation and to show that in the weak noise limit the position of the front can be completely described by the effect of the noise on the first particle