statistical properties of shocks in burgers turbulence we consider the statistical properties of solutions of burgers equation in the limit of vanishing viscosity frac partial partial t u left x t right frac partial partial x left frac 1 2 u left x t right 2 right 0 with gaussian whitenoise initial data this system was originally proposed by burgers 1 as a crude model of hydrodynamic turbulence and more recently by zel dovichet al 12 to describe the evolution of gravitational matter at large spatio temporal scales with shocks playing the role of mass clusters we present here a rigorous proof of the scaling relationp s s 1 2 s 1 wherep s is the cumulative probability distribution of shock strengths we also show that the set of spatial locations of shocks is discrete i e has no accumulation points and establish an upper bound on the tails of the shock strength distribution namely 1 p s exp cs 3 fors 1 our method draws on a remarkable connection existing between the structure of burgers turbulence and classical probabilistic work on the convex envelope of brownian motion and related diffusion processes

vertex based finite volume solution of the two dimensional navier stokes equations in the work the efficiency of the runge kutta scheme and the benefit of convergence acceleration techniques have been utilized the results obtained for turbulent flow past a naca 0012 airfoil have been compared with available numerical and experimental results

periodic solutions for non linear equations of structured populations we show the existence of non trivial periodic solutions for a class of non linear equations model of age structured populations to this aim we use the theory of centre manifold for a class of abstract differential equations introduced by desch and schappacher and show that a hopf bifurcation theorem can be proved for this class of equations finally we consider some special cases and present an algorithm for the determination of the type of bifurcation and show that in the considered cases the bifurcation is always supercritical

behavioral analysis of a nonlinear three staged population model with age size structure we present and analyze a nonlinear egg juvenile adult model in which eggs and juveniles are structured by age while adults by size the model consists of three first order partial differential equations with initial and boundary conditions existence and uniqueness of nonnegative solutions to the state system are investigated by the method of characteristics and a comparison principle extinction or persistence conditions of the population are obtained via the upper lower solution technique our investigation extends some results in the literature

nonlinear age dependent diffusive equations a bifurcation approach the main goal of this paper is the study of the existence and uniqueness of positive solutions of some nonlinear age dependent diffusive models arising from dynamic populations we use a bifurcation method for which it has been necessary to study in detail the linear and eigenvalue problems associated to the nonlinear problem in an appropriate space

stability analysis for equilibria in age specific population dynamics on considere une population de n especes dont les taux de vie dependent de l ge on etudie le comportement de stabilite a l equilibre pour les equations d evolution

superlinear indefinite systems beyond lotka volterra models abstract this paper analyzes the dynamics of a superlinear indefinite parabolic system as a byproduct a number of new results related to population dynamics and economy are obtained among them it is shown that the presence of refuge areas in competitive environments is an optimal mechanism to avoid extinction and that incorporating local symbiosis in competitive environments increases productivity and allows avoiding extinction of the weaker species undoubtedly a paradigm of global markets and possibly of earth biodiversity our analysis combines a series of well known results for systems with some very recent pioneering findings within the context of superlinear indefinite equations

an overview of semi continuity results on the spectral radius and positivity our aim is to provide a novelly comprehensive and unifying approach to showing the continuous dependence of the spectral radius of compact linear operators defined on banach spaces with specific attention to positive operators defined on normal banach spaces and emphasizing that the upper semi continuity generally holds unlike the lower semi continuity

stable coexistence states in the volterra lotka competition model with diffusion we study the existence uniqueness and stability of coexistence states in the lotka volterra model with diffusion for two competing species we assume that the parameters describing the interactio

hopf bifurcation in a size structured population dynamic model with random growth abstract this paper is devoted to the study of a size structured model with ricker type birth function as well as random fluctuation in the growth process the complete model takes the form of a reaction diffusion equation with a nonlinear and nonlocal boundary condition we study some dynamical properties of the model by using the theory of integrated semigroups it is shown that hopf bifurcation occurs at a positive steady state of the model this problem is new and is related to the center manifold theory developed recently in p magal s ruan center manifold theorem for semilinear equations with non dense domain and applications to hopf bifurcation in age structured models mem amer math soc in press for semilinear equation with non densely defined operators

persistence and extinction in continuous age structured population models abstract survival analyses of populations are developed in a continuous process persistence and extinction criteria of a class of continuous age structured population models with separable mortality function and fertility function are established by investigation of asymptotic behaviors of solutions of mckendrick von foerster equation

the concave majorant of brownian motion let s t be a version of the slope at time t of the concave majorant of brownian motion on lbrack 0 infty it is shown that the process s 1 s t t 0 is the inverse of a pure jump process with independent nonstationary increments and that brownian motion can be generated by the latter process and brownian excursions between values of the process at successive jump times as an application the limiting distribution of the l 2 norm of the slope of the concave majorant of the empirical process is derived

on the semigroup for age dependent population dynamics with spatial diffusion abstract we continue a population model with age dependence and spatial diffusion in the semigroup framework in which the assumptions of our previous paper manuscripta math 66 1990 161 180 that birth and death rates are independent of spatial variables are removed the infinitesimal generator is identified and its spectrum studied and accordingly by using a positive semigroup theory we determine its dominant eigenvalue and hence the asymptotic expression is obtained

nonlinear age dependent population dynamics with random diffusion abstract a model of population growth is developed which allows for consideration of both age and spatial effects the mortality and fertility processes of the population are allowed to depend nonlinearly upon the age structure and the population is allowed to diffuse randomly in a spatial region the problem is studied abstractly as an evolution equation in a banach space of density functions the problems investigated are existence and uniquencess of solutions continious dependence upon variations in the mortality and fertility functions and continous dependence upon the initial population distribution

bifurcation of steady state solutions in predator prey and competition systems we discuss steady state solutions of systems of semilinear reaction diffusion equations which model situations in which two interacting species u and v inhabit the same bounded region it is easy to find solutions to the systems such that either u or v is identically zero such solutions correspond to the case where one of the species is extinct by using decoupling and global bifurcation theory techniques we prove the existence of solutions which are positive in both u and v corresponding to the case where the populations can co exist

on positive solutions of some pairs of differential equations this paper is a sequel to the author s paper 6 which studied positive solutions of a pair of differential equations introduced by conway gardner and smoller 4 their equations were equations for population problems in biology their significance is discussed further in 141 the purpose of the present paper is to obtain further results for the conwat gardner smoller equations and to show how the methods in 6 can be modified to decide exactly when the classical predator prey system as in blat and brown 2 has a strictly positive solution we also show how the methods can be applied to a number of other problems in particular our results answer some questions in conway s secondly we show how the asymptotic methods developed in 6 can be used to study the existence and uniqueness of strictly positive solutions of competing species problems our results suggest that this is a much more complicated problem than the predator prey problem we do obtain a condition which is except for a few special cases necessary and sufficient for the existence of a strictly positive solution however the condition is complicated and rather implicit thirdly we prove the uniqueness of the strictly positive solution in the conway smoller system if d is small and n 1 that is the case of ordinary differential equations this partially answers a question in 4 our proof is a local result and can be used in other situations for example it can be used to show that in certain cases of asocial nonlinearities where our notation follows 4 sect 4 there are exactly two strictly positive solutions for small d we also show how our method applies to more general predator prey problems in section 1 we find necessary and sufficient conditions for the existence of strictly positive solutions of predator prey systems in section 2 we study competing species models by iteration and asymptotic methods finally in section 3 we prove our uniqueness results in an appendix we briefly discuss how the methods of section 3 apply to more general models

exponential stabilization of volterra integrodifferential equations in hilbert space we consider exponential stabilization of an abstract linear volterra integrodifferential equation in a hilbert space u e1au t e2 01k t s au s ds f t by a feedback control f t c0u t c1u t with bounded linear operators c0 and c1 of finite rank a is a positive semidefinite self adjoint unbounded operator k is nonnegative convex and decreasing with exponential decay k 0 being finite we introduce a concept of essential growth rate for resolvent operators similar to that for c0 semigroups and prove that the resolvent for the volterra equation has the same essential growth as the solution semigroup to u t e1au t e2k 0 e1u t f t stability can be checked by the location of poles and it is shown that exponential stabilization is possible with a decay rate of at most e2k 0 2e1 we apply our results to two problems concerning stabilization of motion of viscoelastic bodies

bifurcation of time periodic solutions of the mckendrick equations with applications to population dynamics abstract a local two parameter bifurcation theorem concerning the bifurcation from steady states of time periodic solutions of a nonlinear system of partial integro differential equations is proved a hopf bifurcation theorem is derived as a corollary by means of independent and dependent variable changes this theorem is applicable to the general mckendrick equations governing the growth of an age structured population with the added feature here of a possible gestation period the theorem is based on a fredholm theory developed in the paper for the associated linear equations an application is given to an age structured population whose fecundity is density and age dependent and it is shown that for a sufficiently narrow age specific reproductive and resource consumption window steady state instabilities accompanied by sustained time periodic oscillations occur when the birth modulus surpasses a critical value

global bifurcation of positive equilibria in nonlinear population models abstract existence of nontrivial nonnegative equilibrium solutions for age structured population models with nonlinear diffusion is investigated introducing a parameter measuring the intensity of the fertility global bifurcation is shown of a branch of positive equilibrium solutions emanating from the trivial equilibrium moreover for the parameter independent model we establish existence of positive equilibria by means of a fixed point theorem for conical shells

euler and navier stokes solutions for flow over a conical delta wing the simulation of vortical flow about a conical delta wing 14 1 elliptic cone by the euler equations is studied navier stokes solutions are generated for comparison with the inviscid results at the flight condition considered the viscous solutions reveal a large primary leading edge separation vortex and a smaller secondary vortex the viscous solutions are shown to be grid resolved on a coarse grid the euler solutions result in a primary separation vortex but no secondary vortex a comparison of pressure coefficient with the viscous solution shows surprising agreement the agreement is unfortunately fortuitous when the euler equations are solved on the fine viscous grid the large separation vortex is eliminated instead a cross flow shock appears shock curvature introduces sufficient vorticity to produce a small vortex downstream of the shock this inviscid result fundamentally different from the viscous case is a valid solution to the euler equations the coarse grid euler solution is characterized by the production of spurious vorticity and entropy due to numerical error at the leading edge it is neither a valid solution to the euler equations nor a reliable approximation to the real viscous flow the use of a kutta condition is investigated for both the fineand coarse grid euler solutions it is shown that the kutta condition fails to produce a grid independent simulation of the actual viscous flow

global branches of equilibrium solutions of the mckendrick equations for age structured population growth abstract the existence of positive equilibrium solutions of the mckendrick equations for the dynamics of an age structured population is studied as a bifurcation phenomenon using the inherent net reproductive rate n as a bifurcation parameter under only continuity assumptions on the density dependent death and tertility rates it is shown that a global continuum of positive equilibria exists in a certain banach space this continuum connects from the trivial solution at n 1 to the boundary of the domain on which the problem is posed results concerning the spectrum are given in particular some circumstances are described under which positive equilibria exist for all n values greater than the critical value n 1

limit theory for moving averages of random variables with regularly varying tail probabilities let z k infty k infty be iid where the z k s have regularly varying tail probabilities under mild conditions on a real sequence c j j geq 0 the stationary process x n sum infty j 0 c jz n j n geq 1 exists a point process based on x n converges weakly and from this a host of weak limit results for functionals of x n ensue we study sums extremes excedences and first passages as well as behavior of sample covariance functions

stability and regularity results for a size structured population model in the present paper a nonlinear size structured population dynamical model with size and density dependent vital rate functions is considered the linearization about stationary solutions is analyzed by semigroup and spectral methods in particular the spectrally determined growth property of the linearized semigroup is derived from its long term regularity these analytical results make it possible to derive linear stability and instability results under biologically meaningful conditions on the vital rates the principal stability criteria are given in terms of a modified net reproduction rate

structured populations the stabilizing effect of the inflow of newborns from an external source and the net growth rate we investigate the effect of a positive population inflow of individuals from an external source on the dynamical behaviour of certain physiologically structured population models we treat a size structured model with constant inflow and nonlinear birth rate and an age structured model with nonlinear density dependent inflow and linear birth rate analogously to the inherent net reproduction rate we introduce a net growth rate and discuss how this net growth rate can be related to our stability instability conditions

on positive solutions of some pairs of differential equations ii on etudie l existence de solutions strictement positives de systemes predateur proie on demontre l unicite de la solution strictement positive dans le systeme de conway smoller

a nonlinear age dependent model with spatial diffusion abstract the main goal of this paper is the study of the existence and uniqueness of a positive solution for a nonlinear age dependent equation with spatial diffusion for that we mainly use the properties of an eigenvalue problem related to the equation and the sub supersolution method we justify that this method works for this equation in which there is a potential blow up and a nonlocal initial condition

hausdorff dimension of regular points in stochastic burgers flows with levy α stable initial data this paper studies statistical properties of shocks for the inviscid burgers equation with an stable levy motion initial data in the absence of analytic results numerical and computer simulation tools are utilized qualitative and quantitative information on the scaling properties of lagrangian regular points of solutions is obtained and in particular their hausdorff dimension is estimated to be 1 this suggestsa possible extension of ya sinai s result for brownian initial data

boundedness global existence and continuous dependence for nonlinear dynamical systems describing physiologically structured populations abstract the paper is aimed as a contribution to the general theory of nonlinear infinite dimensional dynamical systems describing interacting physiologically structured populations we carry out continuation of local solutions to maximal solutions in a functional analytic setting for maximal solutions we establish global existence via exponential boundedness and by a contraction argument adapted to derive uniform existence time moreover within the setting of dual banach spaces we derive results on continuous dependence with respect to time and initial state to achieve generality the paper is organized top down in the way that we first treat abstract nonlinear dynamical systems under very few but rather strong hypotheses and thereafter work our way down towards verifiable assumptions in terms of more basic biological modelling ingredients that guarantee that the high level hypotheses hold

numerical solution of navier stokes equations for two dimensional viscous compressible flows a finite volume space discretization of viscous fluxes in compressible navier stokes equations is presented to advance the solution in time an explicit five stage runge kutta scheme has been used to accelerate the rate of convergence to steady state local time stepping residual averaging and enthalpy damping have been employed the scheme has been evaluated by solving laminar flow over a iemi infinite flat plate and an naca 0012 air foil using thinlayer approximation it has been observed here that fourthorder artificial dissipation is sufficient for numerical stability the results have been compared with available theoretical and numerical solutions

stable non gaussian random processes stochastic models with infinite variance stable random variables on the real line multivariate stable distributions stable stochastic integrals dependence structures of multivariate stable distributions non linear regression complex stable stochastic integrals and harmonizable processes self similar processes chentsov random fields introduction to sample path properties boundedness continuity and oscillations measurability integrability and absolute continuity boundedness and continuity via metric entropy integral representation historical notes and extensions

the inviscid burgers equation with initial data of brownian type the solutions to burgers equation in the limit of vanishing viscosity are investigated when the initial velocity is a brownian motion or fractional brownian motion function i e a gaussian process with scaling exponent 0 h 1 typea or the derivative thereof with scaling exponent 1 h 0 typeb largesize numerical experiments are performed helped by the fact that the solution is essentially obtained by performing a legendre transform the main result is obtained for typea and concerns the lagrangian functionx a which gives the location at timet 1 of the fluid particle which started at the locationa it is found to be a complete devil s staircase the cumulative probability of lagrangian shock intervals a also the distribution of shock amplitudes follows a a h law for small a the remaining regular lagrangian locations form a cantor set of dimensionh in eulerian coordinates the shock locations are everywhere dense the scaling properties of various statistical quantities are also found heuristic interpretations are provided for some of these results rigorous results for the case of brownian motion are established in a companion paper by ya sinai for typeb initial velocities e g white noise there are very few small shocks and shock locations appear to be isolated finally it is shown that there are universality classes of random but smooth non scaling initial velocities such that the long time large scale behavior is after rescaling the same as for typea orb

the semigroup associated with nonlinear age dependent population dynamics abstract the solutions of the equations of nonlinear age dependent population dynamics may be associated with a strongly continuous semigroup of nonlinear operators in the banach space l 1 0 r n the infinitesimal generator of this nonlinear semigroup is characterized and an exponential representation formula for the nonlinear semigroup is obtained in terms of the infinitesimal generator

endemic thresholds and stability in a class of age structured epidemics an age structured epidemic model is analyzed when the fertility mortality and removal rates depend on age for certain general forms of the force of infection terms endemic threshold criteria are derived and the stability of steady state solutions is determined the relation between age structured models of this type and catalytic curve models of epidemics is derived the possibility of identifying vertically transmitted diseases from the catalytic curve is demonstrated

dynamics of age structured predator prey population models a predator prey model is investigated in which the prey population is assumed to have age structure and is governed by the mckendrick von foerster partial differential equation and the predator population is described by the classical volterra lotka ordinary differential equation quite general hypotheses are assumed for the mortality function the fertility function and the functional responses of predation existence and stability of three biologically meaningful equilibria corresponding to extinction of both species persistence of one species prey and coexistence of two species are studied a particular example and some numerical results are given

irreducible compact operators the purpose of the present paper is to show that a positive compact ideal irreducible operator t in a banach lattice of dimension greater than one has spectral radius r t 0 i e t is not quasi nilpotent it was shown in 6 that there exist positive irreducible operators which are quasi nilpotent and the question under what conditions a positive irreducible operator t satisfies r t 0 has been studied extensively see e g 6 and 7 sect v 6 in particular we mention the ando krieger theorem see e g 9 theorem 136 9 which states that a positive irreducible kernel operator t in a banach function space has r t 0 recently in 8 some special situations are discussed in which a compact positive irreducible operator has a strictly positive spectral radius let l be a real or complex banach lattice by 2 we denote the banach space of all bounded linear operators in l as usual we write s t in l whenever su tu for all o uel given 0 t l the closed ideal j in l is called t invariant if t j j and t is called ideal irreducible if the only closed t invariant ideals are 0 and l see 7 iii 8 for the general theory of banach lattices and positive operators we refer to the books 7 and 9 in the proof of the main result of the paper we will use some properties of the center of a banach lattice we recall some elementary facts let l be a real banach lattice the center z l of l is the subspace of 5 l consisting of all operators z for which there exists 0 2 ir such that f l 2 f for all f l the center has the structure of a vector lattice with nl v n2 u n l u v n2u and ozla zz u lu a nzu for all o u l and all l n2ez l moreover z l is a commutative algebra with respect to composition as multiplication and the identity operator i in l is a strong unit i e for any rc z l there exists 0 2eir such that zi 2i see 9 chap 20 note that if l c k for some compact hausdorff space k then z l can be identified with c k acting on itself by multiplication if l is a dedekind complete banach lattice and 0 u v in l then it is an easy consequence of the freudenthal spectral theorem 4 theorem 40 2 that there exists n z l such that zcv u and o n i however in an arbitrary banach lattice such a 7z need not exist as is shown by the banach lattice l c 0 1 the next lemma shows that under rather mild conditions we can

remarks on semilinear partial functional differential equations with infinite delay abstract in this paper we will be concerned with the abstract semilinear equation x t ax t f t xt t x where a represents the infinitesimal generator of a linear c0 semigroup in a banach space and x denotes the segment of the solution x due to the initial function taken from up to t conditions for existence continuous dependence and regularity of solutions are given which are independent of the topology of the phase space as well as conditions which guarantee the compactness of bounded orbits even if exp at is not compact for t 0

domain perturbation for linear and nonlinear parabolic equations abstract we shall develop a general theory for domain perturbation for linear and nonlinear parabolic equations with measurable coefficients subject to dirichlet boundary conditions we show how solutions of linear and nonlinear parabolic equations behave as a sequence of domains approaches an open set convergence of domains is understood in a very general sense which allows that certain parts of the domains degenerate and are deleted in the limit or that small sets are removed we also consider the periodic problem and establish existence and uniqueness results for periodic solutions for the perturbed problem in a neighborhood of a periodic solution of the original equation the theory can be used for instance to construct domains where a given nonlinear periodic parabolic equation has many periodic solutions

logistic models of structured population growth abstract a general model of structured population dynamics with logistic type nonlinearity is considered the model consists of a semilinear hyperbolic partial differential equation with a linear boundary condition and an initial value the logistic nonlinearity corresponds to increased mortality as the population increases the problem is treated as an abstract semilinear evolution equation sufficient conditions are given for the solutions to converge to an equilibrium solution applications are made to age structured populations size structured populations multispecies structured populations and cell populations

hyperbolic asymptotics in burgers turbulence and extremal processes large time asymptotics of statistical solutionu t x 1 2 of the burgers equation 1 1 is considered where x l x is a stationary zero mean gaussian process depending on a large parameterl 0 so that xi l x sim sigma l eta x l l to infty where sigma l l 2 2 log l 1 2 and x is a given standardized stationary gaussian process we prove that asl the hyperbolicly scaled random fieldsu l2t l2x converge in distribution to a random field with saw tooth trajectories defined by means of a poisson process on the plane related to high fluctuations of x which corresponds to the zero viscosity solutions at the physical level of rigor such asymptotics was considered before by gurbatov malakhov and saichev 1991

the steady states of a non cooperative model of nuclear reactors abstract this paper characterizes the existence of coexistence states in a reaction diffusion model arising in the theory of nuclear reactors from a mathematical point of view the importance of this model relies upon the fact that the associated variational systems are of non cooperative type and consequently the comparison techniques available for cooperative systems fail to work out although in higher spatial dimensions the dynamics of the model might be rather involved by the absence of limitations for the number of steady states we can prove the uniqueness of the steady state in the one dimensional prototype model our results complement and eventually sharpen the findings of arioli g arioli long term dynamics of a reaction diffusion system j differential equations 235 2007 298 307

stability results for a size structured population model with resources dependence and inflow it is undoubted that the survival of individuals of populations is dependent on resources e g foods we formulate a system of integro differential equations to model the dynamics of a size structured and resources dependent population a kind of inflow of newborn individuals from external environment is considered the resource dependence is incorporated through the size growth mortality fertility and feeding rates of the target population the existence of the stationary size distributions are discussed and the linear stability is investigated by means of the semigroup theory and the characteristic equation technique some sufficient conditions for stability instability of stationary states are obtained and two examples and the corresponding simulations are presented

local stability of an age structured population with density dependent fertility and mortality the local stability of an equilibrium population configuration of a nonlinear continuous age structured model with fertility and mortality dependent on total population size is investigated two demographic parameters the marginal birth and death rates which measure the sensitivities of the fertility and mortality of the equilibrium population configuration to changes in population size are introduced it is shown that in certain cases the values of these two parameters completely determine the stability classification while in other cases further information is required examples illustrating these results are presented

on the qualitative behaviour of populations with age specific interactions abstract a mathematical model describing the age dependent dynamics of a population is studied under very general assumptions existence of nonzero equilibrium solutions is proven for some special kind of rate law it is shown that this model can asymptotically be described by means of a simpler system of ordinary differential equations an example is given where the model admits nonconstant time periodic solutions

local and global stability for the solutions of a nonlinear renewal equation existence uniqueness and stability of the stationary solutions of the problem ua a t ut a t m a u a t u a t u 0 t 0 b a u a t u a t da u a 0 u0 a 0 a t 0 with m b and given are investigated together with their ecological implications

the maximum principle and the existence of principal eigenvalues for some linear weighted boundary value problems abstract in this work we deal with the problem of the existence and uniqueness of principal eigenvalues for some linear weighted boundary value problems associated to a general second order uniformly elliptic operator for a large class of sign definited weights we characterize whether the boundary value problem admits a principal eigenvalue or not

periodic mckendrick equations for age structered population growth abstract with the averaged net reproductive rate used as a bifurcation parameter the existence of a local parameterized branch of time periodic solutions of the mckendrick equations is proved under the assumption that the death and fertility rates suffers small amplitude time periodicities the required linear theory is developed and the results are illustrated by means of a simple example in which fertility varies cosinusoidally in time

computation of transonic flows with shock induced separation using algebraic turbulence models a finite volume method based on the nodal point approach was used to develop an algebraic turbulence model for the computation of the transonic flows with shock induced separation three examples were considered using a c type algebraic grid results of the study suggest that the flow between the wall and the minimum velocity line inside the bubble can be treated as laminar and outside the minimum velocity line the flow behaves like attached flow

large deviations estimates in burgers turbulence with stable noise initial data we consider the inviscid burgers equation where the initial datum is given by a stable levy noise the asymptotic behavior of the tail distribution of the solution is described the decay is much faster in the case when the stable noise is completely skewed to the left

structure of shocks in burgers turbulence with levy noise initial data we study the structure of the shocks for the inviscid burgers equation in dimension 1 when the initial velocity is given by levy noise or equivalently when the initial potential is a two sided levy process 0 when 0 is abrupt in the sense of vigon or has bounded variation with lim sup h 0h 2 0 h we prove that the set of points with zero velocity is regenerative and that in the latter case this set is equal to the set of lagrangian regular points which is non empty when 0 is abrupt we show that the shock structure is discrete when 0 is eroded we show that there are no rarefaction intervals

statistical properties of shocks in burgers turbulence ii tail probabilities for velocities shock strengths and rarefaction intervals this paper studies the structure of the random sawtooth profile corresponding to the solution of the inviscid burgers equation with white noise initial data this function consists of a countable sequence of rarefaction waves separated by shocks we are concerned here with calculating the probabilities of rare events associated with the occurrence of very large values of the normalized velocity shock strength and rarefaction intervals we find that these quantities have tail probabilities of the form exp cx3 x 1 this cubic exponential decay of probabilities was conjectured in the companion paper 1 the calculations are done using a representation of the shock strength and length of rarefaction intervals in terms of the statistics of certain conditional diffusion processes

stability and instability of equilibria of an equation of size structured population dynamics in this paper we consider a quasilinear equation with a nonlinear boundary condition modelling the dynamics of a biological population structured by size we suppose vital rates depending on the total population this hypothesis introduces some nonlinearities on the equation and on the boundary condition we study the existence and uniqueness of solution of the initial value problem and the existence of stationary solutions after we calculate the spectrum of the linearization at an equilibrium and we study its local stability

spectral properties of burgers and kpz turbulence this paper presents the higher order spectral densities of non gaussian random fields arising as scaling limits in the burgers and kpz turbulence problems with strongly dependent non gaussian initial conditions

endemic threshold results for an age structured sis epidemic model with periodic parameters abstract the main contribution of this paper is to obtain a threshold value for the existence and uniqueness of a nontrivial endemic periodic solution of an age structured sis epidemic model with periodic parameters under the assumption of the weak ergodicity of a non autonomous lotka mckendrick system we formulate a normalized system for an infected population as an initial boundary value problem of a partial differential equation the existence problem for endemic periodic solutions is reduced to a fixed point problem of a nonlinear integral operator acting on a banach space of locally integrable periodic l 1 valued functions we prove that the spectral radius of the frechet derivative of the integral operator at zero plays the role of a threshold for the existence and uniqueness of a nontrivial fixed point of the operator corresponding to a nontrivial periodic solution of the original differential equation in a weak sense if the malthusian parameter of the host population is equal to zero our threshold value is equal to the well known epidemiological threshold value the basic reproduction number r 0 however if it is not the case then two threshold values are different from each other and we have to pay attention on their actual biological implications

on the stability of stationary age distributions age structured nonlinear single species and predator prey population models are treated and a straightforward relatively simple method is shown to arrive at a condition of stability of stationary solutions