sequentiality monadic second order logic and tree automata abstract given a term rewriting system r and a normalizable term t a redex is needed if in any reduction sequence of t to a normal form this redex will be contracted roughly r is sequential if there is an optimal reduction strategy in which only needed redexes are contracted more generally g huet and j j levy have defined the sequentiality of a predicate p on partially evaluated terms 1991 computational logic essays in honor of alan robinson mit press cambridge ma pp 415 443 we show here that the sequentiality of p is definable in sks the monadic second order logic with k successors provided p is definable in sks we derive several known and new consequences of this remark 1 strong sequentiality as defined by huet and levy of a left linear possibly overlapping rewrite system is decidable 2 nv sequentiality as defined in m oyamaguchi 1993 siam j comput 19 424 437 is decidable even in the case of overlapping rewrite systems 3 sequentiality of any linear shallow rewrite system is decidable then we describe a direct construction of a tree automaton recognizing the set of terms that do have needed redexes which again yields immediate consequences 1 strong sequentiality of possibly overlapping linear rewrite systems is decidable in exptime 2 for strongly sequential rewrite systems needed redexes can be read directly on the automaton

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

long time existence for signed solutions of the heat equation with a noise term let be the circle 0 j with the ends identified we prove long time existence for the following equation

permutable indistinguishability operators perfect vague groups and fuzzy subgroups permutability between t indistinguishability operators is a very interesting property that is related to the compatibility of the operators with algebraic structures it will be shown that the sup t product email protected f of two t indistinguishability operators is also a t indistinguishability operator if and only if e and f are permutable t indistinguishability operators i e email protected email protected e this property will be related to the study of fuzzy subgroups fuzzy normal subgroups and vague groups the aggregation of fuzzy subgroups will also be analyzed

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

inference of fuzzy relational tableaux from fuzzy exemplifications the paper describes a method for deriving fuzzy relational tableaux from fuzzy exemplifications in particular the work by thole zimmerman and zysno 4 is cited as they derive an operator for and using data derived from human subjects the method results in an operator or operators which is a maximal restriction of the operators implied by the data results showing the process are described and a value of and is derived using data presented in 4

a solution to an open problem on the t product of tl fuzzy subgroups many studies have investigated necessary and sufficient conditions whereby fuzzy subgroups of a cartesian product of groups are a cartesian product of fuzzy subgroups chon fuzzy subgroups as products fuzzy sets and systems 141 2004 505 508 found such conditions under minimum operation he pointed out that the problem of finding necessary and sufficient conditions for t fuzzy subgroups under a t norm operation is still open in this study we solve the problem and identify necessary and sufficient conditions under t norm operation

random traveling wave and bifurcations of asymptotic behaviors in the stochastic kpp equation driven by dual noises abstract in this paper we study the influences of dually environmental noises on the traveling wave which develops from the deterministic kpp equation we prove that if the strengths of noises satisfy some condition the solution of the stochastic kpp equation with heaviside initial condition develops a random traveling wave whose wave speed is deterministic and depends on the strengths of noises if the strengths of noises satisfy some other conditions the solution tends to zero as time tends to infinity therefore there exist bifurcations of asymptotic behaviors of solution induced by the strengths of dual noises

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

a collective coordinate framework to study the dynamics of travelling waves in stochastic partial differential equations abstract we propose a formal framework based on collective coordinates to reduce infinite dimensional stochastic partial differential equations spdes with symmetry to a set of finite dimensional stochastic differential equations which describe the shape of the solution and the dynamics along the symmetry group we study spdes arising in population dynamics with multiplicative noise and additive symmetry breaking noise the collective coordinate approach provides a remarkably good quantitative description of the shape of the travelling front as well as its diffusive behaviour which would otherwise only be available through costly computational experiments we corroborate our analytical results with numerical simulations of the full spde

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

last passage percolation and traveling fronts we consider a system of n particles with a stochastic dynamics introduced by brunet and derrida phys rev e 70 016106 2004 the particles can be interpreted as last passage times in directed percolation on 1 n of mean field type the particles remain grouped and move like a traveling front subject to discretization and driven by a random noise as n increases we obtain estimates for the speed of the front and its profile for different laws of the driving noise as shown in brunet and derrida phys rev e 70 016106 2004 the model with gumbel distributed jumps has a simple structure we establish that the scaling limit is a levy process in this case we study other jump distributions we prove a result showing that the limit for large n is stable under small perturbations of the gumbel in the opposite case of bounded jumps a completely different behavior is found where finite size corrections are extremely small

stochastic generalised kpp equations we classify multiplicative white noise perturbations k dw of generalised kpp equations and their effects on deterministic approximate travelling wave solutions by the behaviour of the solutions of the stochastic generalised kpp equations converge to deterministic approximate travelling waves and if being an associated potential energy s a solution of the corresponding classical mechanical equations of newton d being a certain domain in r1 rr then the white noise perturbations essentially destroy the wave structure and force the solutions to die down for the case suppose the existence of the limit we show that there is a residual wave form but propagating at a different speed from that of the unperturbed equations numerical solutions are included and give good agreement with theoretical results

stochastic p d e s arising from the long range contact and long range voter processes a long range contact process and a long range voter process are scaled so that the distance between sites decreases and the number of neighbors of each site increases the approximate densities of occupied sites under suitable tine scaling converge to continuous space time densities which solve stochastic p d e s for the contact process the limiting equation is the kolmogorov petrovskii piscuinov equation driven by branching white noise for the voter process the limiting equation is the heat equation driven by fisher wright white noise

spatial extent of branching brownian motion we study the one dimensional branching brownian motion starting at the origin and investigate the correlation between the rightmost x max geq 0 and leftmost x min leq 0 visited sites up to time t at each time step the existing particles in the system either diffuse with diffusion constant d die with rate a or split into two particles with rate b we focus on the regime b leq a where these two extreme values x max and x min are strongly correlated we show that at large time t the joint probability distribution function pdf of the two extreme points becomes stationary p x y t to infty to p x y our exact results for p x y demonstrate that the correlation between x max and x min is nonzero even in the stationary state from this joint pdf we compute exactly the stationary pdf p zeta of the dimensionless span zeta x max x min sqrt d b which is the distance between the rightmost and leftmost visited sites this span distribution is characterized by a linear behavior p zeta sim frac 1 2 left 1 delta right zeta for small spans with delta left frac a b 1 right in the critical case delta 0 this distribution has a non trivial power law tail p zeta sim 8 pi sqrt 3 zeta 3 for large spans on the other hand in the subcritical case delta 0 we show that the span distribution decays exponentially as p zeta sim a 2 2 zeta exp left sqrt delta zeta right for large spans where a is a non trivial function of delta which we compute exactly we show that these asymptotic behaviors carry the signatures of the correlation between x max and x min finally we verify our results via direct monte carlo simulations

travelling waves for reaction diffusion equations forced by translation invariant noise abstract inspired by applications we consider reaction diffusion equations on r that are stochastically forced by a small multiplicative noise term that is white in time coloured in space and invariant under translations we show how these equations can be understood as a stochastic partial differential equation spde forced by a cylindrical q wiener process and subsequently explain how to study stochastic travelling waves in this setting in particular we generalize the phase tracking framework that was developed in hamster and hupkes 2018 2019 for noise processes driven by a single brownian motion the main focus lies on explaining how this framework naturally leads to long term approximations for the stochastic wave profile and speed we illustrate our approach by two fully worked out examples which highlight the predictive power of our expansions

the scaling limit for a stochastic pde and the separation of phases we investigate the problem of singular perturbation for a reaction diffusion equation with additive noise or a stochastic partial differential equation of ginzburg landau type under the situation that the reaction term is determined by a potential with double wells of equal depth as the parameter e the temperature of the system tends to 0 the solution converges to one of the two stable phases and consequently the phase separation is formed in the limit we derive a stochastic differential equation which describes the random movement of the phase separation point the proof consists of two main steps we show that the solution stays near a manifoldm e of minimal energy configurations based on a lyapunov type argument then the limit equation is identified by introducing a nice coordinate system in a neighborhood ofm e

nv sequentiality a decidable condition for call by need computations in term rewriting systems in 1979 huet and levy introduced the class of sequential term rewriting systems in which call by need computations are possible without look ahead and defined the subclass called strongly sequential systems for which needed redexes in a given term are effectively found chapter in computational logic essays in honor of alan robinson j l lassez and g plotkin eds mit press cambridge ma 1991 this paper introduces a larger subclass that is a natural extension of strong sequentiality and is based on the analysis of both the left hand sides and part of the right hand sides i e the nonvariable parts of systems whereas strong sequentiality is based on the analysis of left hand sides alone this new sequentiality is called nv sequentiality it is shown that i the class of nv sequential systems properly includes the class of strongly sequential systems ii there exists an algorithm for finding needed redexes for a given term when a system is nv sequential and iii it is decidable whether a

the stable manifold theorem for stochastic differential equations type equations are treated starting with the existence of a stochastic flow for a sde we introduce the notion of a hyperbolic stationary trajectory we prove the existence of invariant random stable and unstable manifolds in the neighborhood of the hyperbolic stationary solution for stratonovich sdes the stable and unstable manifolds are dynamically characterized using forward and backward solutions of the anticipating sde the proof of the stable manifold theorem is based on ruelle oseledec multiplicative ergodic theory

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

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

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

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

a characterization of fuzzy subgroups abstract an ordinary subgroup of a group g is 1 a subset of g 2 closed under the group operation in a fuzzy subgroup it is precisely these two notions that lose their deterministic character a fuzzy subgroup of a group g associates with each group element a number the larger the number the more certainly that element belongs to the fuzzy subgroup the closure property is captured by the inequality x y t x y in a rosenfeld s original definition t was the function minimum however any t norm t provides a meaningful generalization of the closure property two classes of fuzzy subgroups are investigated the fuzzy subgroups in one class are subgroup generated those in the other are function generated each fuzzy subgroup in these classes satisfies the above inequality with t given by t a b max a b 1 0 while the two classes look different each fuzzy subgroup in either is isomorphic to one in the other it is shown that a fuzzy subgroup satisfies the above inequality with t minimum if and only if it is subgroup generated of a very special type finally these notions are applied to some abstract pattern recognition problems

context sensitive rewriting strategies contex sensitive rewriting is a simple rewriting restriction which is formalized by imposing fixed restrictions on replacements such a restriction is given on a purely syntactic basis it is given on the arguments of symbols of the signature and inductively extended to arbitrary positions of terms built from those symbols the termination behavior is not only preserved but usually improved and several methods have been developed to formally prove it in this paper we investigate the definition properties and use of context sensitive rewriting strategies i e particular fixed sequences of context sensitive rewriting steps we study how to define them in order to obtain efficient computations and to ensure that context sensitive computations terminate whenever possible we give conditions enabling the use of these strategies for root normalization normalization and infinitary normalization we show that this theory is suitable for formalizing the definition and analysis of real computational strategies which are used in programming languages such as obj or elan

ito versus stratonovich a survey is given of the facts and fancies concerning the nonlinear langevin or ito equation actually it is merely a pre equation which becomes an equation when an interpretation rule is added the rules of ito and stratonovich differ but both are mathematically consistent and therefore equally admissible conventions the reason why they seem to lead to physical differences is that the langevin approach used to arrive at the equation involves a tacit assumption for systems with external noise this assumption can be justified and it is then clear that the stratonovich rule applies systems with internal noise however can only be properly described by a master equation and the ito stratonovich controversy never enters afterward one is free to model the resulting fluctuations either with an ito or a stratonovich scheme but that does not lead to any new information

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

computing stochastic traveling waves the aim of this paper is to investigate new numerical methods for computing traveling wave solutions and new ways for estimating characteristic properties such as wave speed for stochastically forced partial differential equations as a particular example we consider the nagumo equation with multiplicative noise which we mainly consider in the stratonovich sense a standard approach for determining the position and hence speed of a wave is to compute the evolution of a level set we compare this approach against an alternative where the wave position is found by minimizing the l 2 norm against a fixed reference profile this approach can be used to freeze or stop the wave and obtain a stochastic partial differential algebraic equation that we then discretize and solve although attractive because it leads to a smaller domain size it can be numerically unstable due to large convection terms we compare numerically the different approaches for estimating the wave speed we use these techniques to investigate the effect of both ito and stratonovich noise on the nagumo equation as correlation length and noise intensity increase

a variational principle for kpp front speeds in temporally random shear flows we establish the variational principle of kolmogorov petrovsky piskunov kpp front speeds in temporally random shear flows with sufficiently decaying correlations a key quantity in the variational principle is the almost sure lyapunov exponent of a heat operator with random potential to prove the variational principle we use the comparison principle of solutions the path integral representation of solutions and large deviation estimates of the associated stochastic flows the variational principle then allows us to analytically bound the front speeds the speed bounds imply the linear growth law in the regime of large root mean square shear amplitude at any fixed temporal correlation length and the sublinear growth law if the temporal decorrelation is also large enough the so called bending phenomenon

asymptotic properties of the solutions to stochastic kpp equations a reduction method is used to prove existence and uniqueness of strong solutions to stochastic kpp equations where the initial condition may be anticipating the asymptotic behavior of the solution for large time and space and the random travelling waves are then studied under two different basic assumptions

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

kpp equation and supercritical branching brownian motion in the subcritical speed area application to spatial trees if r t is the position of the rightmost particle at time t in a one dimensional branching brownian motion u t x p r t x is a solution of kpp equation frac partial u partial t frac 1 2 frac partial 2 u partial x 2 f u where f u 1 u g 1 u g is the generating function of the reproduction law and the inverse of the mean lifetime if m g 1 1 and g 0 0 it is known that frac r t t xrightarrow p c 0 sqrt 2a m 1 text when t to infty for the general kpp equation we show limit theorems for u t ct c c 0 t large deviations for r t and probabilities of presence of particles for the branching process are deduced where z t denotes the random point measure of particles living at time t and a yaglom type theorem is proved the conditional distribution of the spatial tree given z t ct 0 is studied in the limit as t

brunet derrida particle systems free boundary problems and wiener hopf equations we consider a branching selection system in r with n particles which give birth independently at rate 1 and where after each birth the leftmost particle is erased keeping the number of particles constant we show that as n the empirical measure process associated to the system converges in distribution to a deterministic measure valued process whose densities solve a free boundary integrodifferential equation we also show that this equation has a unique traveling wave solution traveling at speed c or no such solution depending on whether c a or c a where a is the asymptotic speed of the branching random walk obtained by ignoring the removal of the leftmost particles in our process the traveling wave solutions correspond to solutions of wiener hopf equations

brunet derrida behavior of branching selection particle systems on the line we consider a class of branching selection particle systems on mathbb r similar to the one considered by e brunet and b derrida in their 1997 paper shift in the velocity of a front due to a cutoff based on numerical simulations and heuristic arguments brunet and derrida showed that as the population size n of the particle system goes to infinity the asymptotic velocity of the system converges to a limiting value at the unexpectedly slow rate log n 2 in this paper we give a rigorous mathematical proof of this fact for the class of particle systems we consider the proof makes use of ideas and results by r pemantle and by n gantert y hu and z shi and relies on a comparison of the particle system with a family of n independent branching random walks killed below a linear space time barrier

on the nonlinear diffusion equation of kolmogorov petrovsky and piscounov abstract recently bramson proved a theorem that classifies the initial data under which solutions of the k p p equation converge to the appropriate travelling waves in this paper a simplified proof is given by using maximal principles instead of his brownian motion approach the regularity condition on the forcing term is also weakened

fuzzy subgroups as products we find necessary and sufficient conditions for a fuzzy subgroup of a cartesian product of groups to be a cartesian product of fuzzy subgroups

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

a sequential reduction strategy kennaway proved the remarkable result that every almost orthogonal term rewriting system admits a computable sequential normalizing reduction strategy in this paper we present a computable sequential reduction strategy similar in scope but simpler and more general our strategy can be thought of as an outerrnost fair like strategy that is allowed to be unfair to some redex of a term when contracting the redex is useless for the normalization of the term unlike the strategy of kennaway our strategy does not rely on syntactic restrictions that imply confluence on the contrary it can easily be applied to any term rewriting system and we show that the class of term rewriting systems for which our strategy is normalizing properly includes all almost orthogonal systems our strategy is more versatile in case of almost orthogonal term rewriting systems it can be used to detect certain cases of non termination our normalization proof is more accessible than kennaway s we also show that our sequential strategy sometimes succeeds where the parallel outermost strategy fails

products of fuzzy subgroups fuzzy subgroups are different from ordinary subgroups in that one cannot tell with certainty which group elements belong and which do not of course this requires an appropriate modification of the closure property which takes the form of an inequality in this paper a study of products of fuzzy subgroups is begun some results are expected the product of fuzzy subgroups is a fuzzy subgroup products with the same but permuted factors are isomorphic and there is a natural isomorphism from any factor into the product functions called t norms are used to construct products of fuzzy subgroups different t norms yielding different products since a fuzzy subgroup must satisfy an inequality generalizing closure and the strength of that inequality is important some results relate the t norm used in forming the product to the strength of the ensuing inequality when can complicated fuzzy subgroups be expressed as products of simple ones this question is addressed for fuzzy subgroups of finite abelian groups

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