a nonlinear functional for general scalar hyperbolic conservation laws abstract a generalized entropy functional was introduced in t p liu t yang a new entropy functional for scalar conservation laws comm pure appl math 52 1999 1427 1442 for the scalar hyperbolic conservation laws with convex flux function this functional was crucially used in the functional approach to the l 1 stability study on the system of hyperbolic conservation laws when each characteristic field is either genuinely nonlinear or linearly degenerate however how to construct the generalized entropy functional for scalar conservation laws with general flux and then how to apply the functional approach to the l 1 study on general systems are still open in this paper we construct a new nonlinear functional which gives some partial answer to this question and we expect the analysis will shed some light on the future investigation in this direction

vanishing viscosity solutions of nonlinear hyperbolic systems we consider the cauchy problem for a strictly hyperbolic n n system in one space dimension ut a u ux 0 assuming that the initial data have small total variation we show that the solutions of the viscous approximations ut a u ux euxx are defined globally in time and satisfy uniform bv estimates independent of e moreover they depend continuously on the initial data in the l 1 distance with a lipschitz constant independent of t e letting e 0 these viscous solutions converge to a unique limit depending lipschitz continuously on the initial data in the conservative case where a df is the jacobian

traveling wave solutions in a diffusive system with two preys and one predator abstract this paper is concerned with the traveling wave solutions in a diffusive system with two preys and one predator by constructing upper and lower solutions the existence of nontrivial traveling wave solutions is established the asymptotic behavior of traveling wave solutions is also confirmed by combining the asymptotic spreading with the contracting rectangles applying the theory of asymptotic spreading the nonexistence of traveling wave solutions is proved

neural associative memory and the willshaw palm probability distribution previous asymptotic analyses of binary neural associative networks of willshaw or steinbuch type relied on a binomial approximation of the neurons dendritic potentials this approximation has been proven to be good only if the stored patterns are extremely sparse for example when the mean number of active units k per pattern vector scales with the logarithm of the vector size n recent promising results concerning storage capacity and retrieval efficiency for larger pattern activities k log n have been doubted because here the binomial approximation can lead to a massive overestimation of performance in this work i compute and characterize the exact willshaw palm distribution of the dendritic potentials for hetero association auto association and fixed and random pattern activity comparing the raw and central moments of the willshaw palm distribution to the moments of the corresponding binomial probability reveals that asymptotically the binomial approximation becomes exact for almost any su

existence and qualitative properties of travelling waves for an epidemiological model with mutations in this article we are interested in a non monotone system of logistic reaction diffusion equations this system of equations models an epidemics where two types of pathogens are competing and a mutation can change one type into the other with a certain rate we show the existence of minimal speed travelling waves that are usually non monotonic we then provide a description of the shape of those constructed travelling waves and relate them to some fisher kpp fronts with non minimal speed

memory capacity for sequences in a recurrent network with biological constraints the ca3 region of the hippocampus is a recurrent neural network that is essential for the storage and replay of sequences of patterns that represent behavioral events here we present a theoretical framework to calculate a sparsely connected network s capacity to store such sequences as in ca3 only a limited subset of neurons in the network is active at any one time pattern retrieval is subject to error and the resources for plasticity are limited our analysis combines an analytical mean field approach stochastic dynamics and cellular simulations of a time discrete mcculloch pitts network with binary synapses to maximize the number of sequences that can be stored in the network we concurrently optimize the number of active neurons that is pattern size and the firing threshold we find that for one step associations i e minimal sequences the optimal pattern size is inversely proportional to the mean connectivity c whereas the optimal firing threshold is independent of the connectivity if the number of synapses per neuron is fixed the maximum number p of stored sequences in a sufficiently large nonmodular network is independent of its number n of cells on the other hand if the number of synapses scales as the network size to the power of 3 2 the number of sequences p is proportional to n in other words sequential memory is scalable furthermore we find that there is an optimal ratio r between silent and nonsilent synapses at which the storage capacity p c 1 r n assumes a maximum for long sequences the capacity of sequential memory is about one order of magnitude below the capacity for minimal sequences but otherwise behaves similar to the case of minimal sequences in a biologically inspired scenario the information content per synapse is far below theoretical optimality suggesting that the brain trades off error tolerance against information content in encoding sequential memories

error bounds for the methods of glimm godunov and leveque the expected error in l 1 r at time t for glimm s scheme when applied to a scalar conservation law is bounded by left h frac 2 sqrt 3 left frac h delta t right 1 2 right left u 0 right bv bf r where h is the mesh size and delta t is the time step we show that the error in godunov s scheme is bounded by the same expression and use this result to analyze a variant of a large time step numerical method of leveque that may be viewed as a combination of godunov s scheme and a method of dafermos

optimised network for sparsely coded patterns the performance of attractor neural networks storing sparsely coded patterns has been shown to be greatly improved on shifting from the 1 1 representation of neural states to the 0 1 representation here the authors show that when this shift is considered as a special case of the transformation of the dynamical variables which depends on a continuous parameter the value of the parameter can be chosen to improve the performance of the network even further for every value of the bias in the patterns

spreading speeds of a lotka volterra predator prey system the role of the predator abstract this paper is concerned with the spreading speeds of a modified lotka volterra predator prey system the intention is to describe the propagation mode whereby the two species coinvade a new habitat under some assumptions it is proved that the spreading speed of the prey is slower than in the case where the predator vanishes and the density of the prey on the coexistence domain is also smaller compared with the case where interspecies actions disappear therefore the predator has a negative effect on the evolution of the prey by slowing the spreading speed and decreasing the population density of the prey

well posedness theory for hyperbolic conservation laws the paper presents a well posedness theory for the initial value problem for a general system of hyperbolic conservation laws we will start with the refinement of glimm s existence theory and discuss the principle of nonlinear through wave tracing our main goal is to introduce a nonlinear functional for two solutions with the property that it is equivalent to the l1 x distance between the two solutions and is time decreasing moreover the functional is constructed explicitly in terms of the wave patterns of the solutions through the nonlinear superposition it consists of a linear term measuring the l1 x distance a quadratic term measuring the coupling of waves and distance and a generalized entropy functional 1999 john wiley sons inc

storage capacity of neural networks effect of the fluctuations of the number of active neurons per memory the storage capacity in an attractor neural network with excitatory couplings is shown to depend not only on the fraction of active neurons per pattern or coding rate but also on the fluctuations around this value in the thermodynamical limit the capacity is calculated in the case of exactly the same number of active neurons in every pattern for every coding level the capacity is increased with respect to the case of random patterns results are supported by numerical simulations done with an exhaustive search algorithm and partly solve in the sparse coding limit the paradox of the discrepancy of the capacity of the willshaw model with optimal capacity

systems of hyperbolic conservation laws with a resonant moving source in this paper we study the stability of a single transonic shock wave solution to the hyperbolic conservation laws with a resonant moving source compared with the previous results w c lien hyperbolic conservation laws with a moving source comm pure appl math 52 9 1999 1075 1098 t p liu nonlinear stability and instability of transonic flows through a nozzle comm math phys 83 2 1982 243 260 on this stability problem in this paper the transonic ith shock is assumed to be relatively strong and stable in the sense of majda then the framework of m lewicka l1 stability of patterns of non interacting large shock waves indiana univ math j 49 4 2000 1515 1537 m lewicka stability conditions for patterns of noninteracting large shock waves siam j math anal 32 5 2001 1094 1116 electronic can be applied a new criterion is obtained to test whether such a shock is time asymptotically stable or not and by constructing the liu yang functional one can prove the l1 stability of the shock under the stability condition this is an extension of the result s y ha t yang l1 stability for systems of hyperbolic conservation laws with a resonant moving source siam j math anal 34 5 2003 1226 1251 electronic w c lien hyperbolic conservation laws with a moving source comm pure appl math 52 9 1999 1075 1098 to a more general case

invasion traveling wave solutions of a predator prey system abstract this paper is concerned with the traveling wave solutions of a predator prey system in which intra specific conflictions are involved by using the comparison principle and the asymptotic speed of spreading the existence nonexistence and minimal wave speed of non negative traveling wave solutions are established such a traveling wave solution can model the spatial temporal process where the predator invades the territory of the prey and they eventually coexist

traveling waves in diffusive predator prey equations periodic orbits and point to periodic heteroclinic orbits we demonstrate the existence of periodic traveling wave train and traveling front solutions for a diffusive predator prey system the analysis is in the three dimensional phase space of the nonlinear ordinary differential equation system given by the diffusive predator prey system in the traveling wave variable the analysis shows the existence of periodic orbits heteroclinic orbits and a heteroclinic connection of a point and a periodic orbit the proof uses shooting techniques invariant manifold theory and the qualitative theory of ordinary differential equations

stability of approximate solutions constructed by the wave front tracking method abstract we prove the stability of approximate solutions constructed by the wave front tracking method for the cauchy problem of scalar conservation laws the ideas of the proof are based upon an application of the proof technique of the uniqueness theorem to the entropy solution given by kru kov and a local generalization of the method by liu and yang

existence and uniqueness of entropy solution to initial boundary value problem for the inviscid burgers equation this paper is concerned with the existence and uniqueness of the entropy solution to the initial boundary value problem for the inviscid burgers equation to apply the method of vanishing viscosity to study the existence of the entropy solution we first introduce the initial boundary value problem for the viscous burgers equation and as in evans 1998 partial differential equations providence ri american mathematical society and hopf 1950 commun pure appl math 3 201 30 give the formula of the corresponding viscosity solutions by hopf cole transformation secondly we prove the convergence of the viscosity solution sequences and verify that the limiting function is an entropy solution finally we give an example to show how our main result can be applied to solve the initial boundary value problem for the burgers equation

on nonlocal monotone difference schemes for scalar conservation laws analyse d erreur d une classe de methodes aux differences finies monotones pour la solution approchee des lois de conservation hyperboliques

equidistributing principles in moving finite element methods abstract recently miller and his co workers proposed a moving finite element method based on a least squares principle this was followed by a similar method by the present authors using a petrov galerkin approach in this paper the two methods are compared in particular it is shown that both methods move their nodes according to an approximate equidistributing principle this observation leads to a criterion for the placement of the nodes it is also shown that the penalty function designed by miller may also be used with the petrov galerkin method finally numerical examples are given illustrating the performance of the two methods

self adjusting grid methods for one dimensional hyperbolic conservation laws it is shown how to automatically adjust the grid to follow the dynamics of the numerical solution of hyperbolic conservation laws the grid motion is determined by averaging the local characteristic velocities of the equations with respect to the amplitudes of the signals the resulting algorithm is a simple extension of many currently popular godunov type methods computer codes using one of these methods can be easily modified to add the moving mesh as an option numerical examples are given that illustrate the improved accuracy of godunov s and roe s methods on a self adjusting mesh

a moving mesh numerical method for hyperbolic conservation laws we show that the possibly discontinuous solution of a scalar conservation law in one space dimension may be approximated in l1 r to within o n 2 by a piecewise linear function with o fn nodes the nodes are moved according to the method of characteristics we also show that a previous method of dafermos which uses piecewise constant approxima tions is accurate to o n 1 these numerical methods for conservation laws are the first to have proven convergence rates of greater than o fn 1 2 1 introduction it is well known that the solution of the hyperbolic conservation law

transient l1 error estimates for well balanced schemes on non resonant scalar balance laws abstract the ability of well balanced wb schemes to capture very accurately steady state regimes of non resonant hyperbolic systems of balance laws has been thoroughly illustrated since its introduction by greenberg and leroux 1996 15 see also the anterior wb glimm scheme in e 1992 8 this paper aims at showing by means of rigorous c t 0 l x 1 estimates that these schemes deliver an increased accuracy in transient regimes too namely after explaining that for the vast majority of non resonant scalar balance laws the c t 0 l x 1 error of conventional fractional step tang and teng 1995 45 numerical approximations grows exponentially in time like exp max g t x as a consequence of the use of gronwall s lemma it is shown that wb schemes involving an exact riemann solver suffer from a much smaller error amplification thanks to strict hyperbolicity their error grows at most only linearly in time see also layton 1984 30 numerical results on several test cases of increasing difficulty including the classical leveque yee s benchmark problem leveque and yee 1990 34 in the non stiff case confirm the analysis

closed form expressions for the moments of the binomial probability distribution this work develops closed form expressions for the raw and central moments of the binomial probability distribution for this i first derive a recursive formula for the raw moments from the moment generating function then it is shown that the recursion involved is essentially the same as for the stirling numbers of the second kind from this fact it is then possible to derive the closed formulae finally i discuss an application of these formulae to the analysis of neural associative memory

the moving grid method for nonlinear hyperbolic conservation laws numerically moving grids are often found to improve the resolution of approximate solutions to nonlinear hyperbolic conservation laws in this paper we show how to modify many standard numerical flux functions to incorporate this idea a generalized godunov type scheme is shown in the scalar case to converge to the entropy satisfying solution as the mesh size tends to zero a courant number of one half is found necessary in general and proved to be sufficient to obtain convergence convergence for methods based upon any two point conservative and monotone numerical flux function is also shown in the scalar case

existence of multidimensional travelling wave solutions of an initial boundary value problem on considere le probleme suivant u t u xx u yy f u u x 0 t u x l t 0 u x y 0 u 0 x y ou x r 1 0 y l et f u u u 1 u avec 0 1 2

moving finite elements i we present new and general numerical methods for dealing with problems whose solutions develop sharp transition layers or near shocks these methods allow many nodes automatically to concentrate in the critical regions and move with them for clarity of exposition we concentrate on the space of piecewise linear functions with movable nodes with burgers equation as our test equation but the generalization to much more general spaces and equations including even certain previous moving vorticity blobs of the first author and s doss for the navier stokes equations becomes clear in this paper we present the theoretical and computational details of our scheme along with computational trial runs for burgers equation showing that the nodes do concentrate sharply and move with the shocks as desired the conclusiveness of these preliminary numerical trials is marred somewhat by the fact that we never successfully debugged a newton s method for our implicit stiff ode solver and were thus limited to ver

the existence of periodic travelling waves for singularly perturbed predator prey equations via the conley index this paper is concerned with the existence of periodic travelling wave solutions of reactiondiffusion equations which arise as models of predator prey interactions in mathematical ecology the relevant equations are somewhat similar to the fitz hugh nagumo equations and our results apply to this latter system as well in studying the fitz hugh nagumo equations conley 3 i initiated a new topological approach to this problem his method depends on the construction of an isolating neighborhood cf section 2a about a singular periodic solution associated with such a neighborhood is a topological invariant the conley index which when nontrivial implies the existence of a smooth solution near the singular one the new aspects of the present discussion are in the method used to construct the isolating neighborhood in ifi4 and in the computation of the index with regard to the isolating neighborhood our construction is global in that a parameter taken to be small in conley s equations is now allowed to assume large values in order to compute the index we continue our equations to those studied by conley actually we go one step further and continue conley s equations to the van der pol equations crossed with a pair of linear equations admitting a repelling critical point the index for this latter system is easily computed and turns out to be non trivial it follows from the invariance of the index under continuation 21 that the desired

storage capacity of a diluted neural network with ising couplings the capacity for storing random patterns in a diluted neural network is determined following the method of gardner 1988 the non zero coupling coefficients are restricted to take on the ising values or 1 different degrees of dilution are considered the maximum values alpha c 1 17 for the storage capacity is obtained when each neuron is connected to 63 of the other neurons

stability of travelling wave solutions of diffusive predator prey systems the stability of travelling wave solutions of singularly perturbed diffusive predator prey systems is proved by showing that the linearized operator about such a solution has no unstable spectrum and that the translation eigenvalue at a 0 is simple the proof illustrates the application of some recently developed geometric and topological methods for counting eigenvalues

on the stability functional for conservation laws this note is devoted to the explicit construction of a functional defined on all pairs of l 1 functions with small total variation which is equivalent to the l 1 distance and non increasing along the trajectories of a given system of conservation laws the present definition of this functional does not need any construction of approximate solutions

precise capacity analysis in binary networks with multiple coding level inputs we compute retrieval probabilities as a function of pattern age for networks with binary neurons and synapses updated with the simple hebbian learning model studied in amit and fusi 1994 the analysis depends on choosing a neural threshold that enables patterns to stabilize in the neural dynamics in contrast to most earlier work where selective neurons for each pattern are drawn independently with fixed probability f here we analyze the situation where f is drawn from some distribution on a range of coding levels in order to set a workable threshold in this setting it is necessary to introduce a simple inhibition in the neural dynamics whose magnitude depends on the total activity of the network proper choice of the threshold depends on the value of the covariances between the synapses for which we provide an explicit formula retrieval probabilities depend on the distribution of the fields induced by a learned pattern we show that the field induced by the first learned pattern evolves as a markov chain during subsequent learning epochs leading to a recursive formula for the distribution alternatively the distribution can be computed using a normal approximation which involves the value of the synaptic covariances capacity is computed as the sum of the retrival probabilities over all ages we show through simulation that the chosen threshold enables retrieval with asynchronous dynamics even in the presence of significant noise in the initial state of the pattern the computed probabilities with both methods are shown to be very close to probabilities estimated from simulation the analysis is extended to randomly connected networks

optimal storage properties of neural network models the authors calculate the number p alpha n of random n bit patterns that an optimal neural network can store allowing a given fraction f of bit errors and with the condition that each right bit is stabilised by a local field at least equal to a parameter k for each value of alpha and k there is a minimum fraction fmin of wrong bits they find a critical line alpha c k with alpha c 0 2 the minimum fraction of wrong bits vanishes for alpha alpha c k the calculations are done using a saddle point method and the order parameters at the saddle point are assumed to be replica symmetric this solution is locally stable in a finite region of the k alpha plane including the line alpha c k but there is a line above which the solution becomes unstable and replica symmetry must be broken

the space of interactions in neural network models the typical fraction of the space of interactions between each pair of n ising spins which solve the problem of storing a given set of p random patterns as n bit spin configurations is considered the volume is calculated explicitly as a function of the storage ratio alpha p n of the value kappa 0 of the product of the spin and the magnetic field at each site and of the magnetisation m here m may vary between 0 no correlation and 1 completely correlated the capacity increases with the correlation between patterns from alpha 2 for correlated patterns with kappa 0 and tends to infinity as m tends to 1 the calculations use a saddle point method and the order parameters at the saddle point are assumed to be replica symmetric this solution is shown to be locally stable a local iterative learning algorithm for updating the interactions is given which will converge to a solution of given kappa provided such solutions exist

a new nonlinear functional for general scalar conservation laws abstract the approach based on the construction of some nonlinear functionals was proved to be robust in the study of the well posedness theories of hyperbolic conservation laws especially in one space dimensional case in particular a generalized entropy functional was constructed in t p liu t yang a new entropy functional for scalar conservation laws comm pure appl math 52 1999 1427 1442 for the l 1 stability of weak solutions however this generalized functional is so far only defined for scalar equations with convex flux function in this paper we introduce a new nonlinear functional which is motivated by the new glimm functional introduced in j l hua z h jiang t yang a new glimm functional and convergence rate of glimm scheme for general systems of hyperbolic conservation laws preprint for general scalar conservation laws this functional improves the one given in h x liu t yang a nonlinear functional for general scalar hyperbolic conservation laws j differential equations 235 2 2007 658 667 and it can be viewed as a better attempt for the generalized entropy functional for general equations

a stable adaptive numerical scheme for hyperbolic conservation laws a new adaptive finite difference scheme for scalar hyperbolic conservation laws is introduced a key aspect of the method is a new automatic mesh selection algorithm for problems with shocks we show that the scheme is l 1 stable in the sense of kuznetsov and that it generates convergent approximations for linear problems numerical evidence is presented that indicates that if an error of size varepsilon is required our scheme takes at most o varepsilon 3 operations standard monotone difference schemes can take up to o varepsilon 4 calculations for the same problems

invariant manifolds and singularly perturbed boundary value problems boundary value problems for singularly perturbed equations often have singular solutions with internal and boundary layers this paper addresses the question of when singular solutions to such prob

a new entropy functional for a scalar conservation law in this paper we introduce a new entropy functional for a scalar convex conservation law that generalizes the traditional concept of entropy of the second law of thermodynamics the generalization has two aspects the new entropy functional is defined not for one but for two solutions it is defined in terms of the l1 distance between the two solutions as well as the variations of each separate solution in addition it is decreasing in time even when the solutions contain no shocks and is therefore stronger than the traditional entropy even in the case when one of the solutions is zero 1999 john wiley sons inc

stability of the travelling wave solution of the fitzhugh nagumo system on demontre la stabilite relative au systeme complet d equations aux derivees partielles de ces ondes de propagation

numerical approximations to nonlinear conservation laws with locally varying time and space grids an explicit time differencing technique is introduced to approximate nonlinear conservation laws this differencing technique links together an arbitrary number of space regimes containing fine and coarse time increments previous stability requirements i e the cfl condition placed a global bound on the size of the time increments for scalar monotone approximations in one space dimension using this variable step time differencing convergence to the correct physical solution is proven given only a local cfl condition

large time step shock capturing techniques for scalar conservation laws for a scalar conservation law u t f u x with f of constant sign the first order upwind difference scheme is a special case of godonov s method the method is equivalent to solving a sequence of riemann problems at each step and averaging the resulting solution over each cell in order to obtain the numerical solution at the next time level the difference scheme is stable and the solutions to the associated sequence of riemann problems do not interact provided the courant number nu is less than 1 by allowing and explicitly handling such interactions it is possible to obtain a generalized method which is stable for nu much larger than 1 in many cases the resulting solution is considerably more accurate than solutions obtained by other numerical methods in particular shocks can be correctly computed with virtually no smearing the generalized method is rather unorthodox and still has some problems associated with it nonetheless preliminary results are quite encouraging

capacity of neural networks with discrete synaptic couplings the authors study the optimal storage capacity of neural networks with discrete local constraints on the synaptic couplings jij models with such constraints include those with binary couplings jij or 1 or jij 0 1 quantised couplings with larger synaptic range e g jij or 1 l or 2 l or 1 and in the limit continuous couplings confined to the hypercube mod jij mod or 1 box confinement they find that the optimal storage capacity alpha kappa is best determined by the vanishing of a suitably defined entropy as calculated in the replica symmetric approximation they also extend their results to cases with biased memories and make contact with sparse coding models

existence of travelling wave solutions of predator prey systems via the connection index we consider a reaction diffusion system which arises as a model of predator prey interactions in mathematical ecology the system admits four constant equilibria two of which are stable the main result is an existence theorem for travelling wave solutions which connect the two stable equilibria most previous efforts with regard to travelling waves for systems employ singular perturbations to locate the connecting orbit the new aspect of the present discussion lies in devising some novel methods for locating the connecting solution in phase space which do not require that the equations be singularly perturbed in certain cases the main result holds for an arbitrary choice of diffusion coefficients

associative memory on the puzzling sparse coding limit recent studies of the information capacity in a sparsely coded memory net has led to some contradictory results in the willshaw model where the couplings are binary 0 or 1 the maximal quantity of information that can be stored is 1n 2 approximately 0 69 bits per synapse on the other hand a calculation a la gardner 1988 for 0 1 couplings gives an upper bound for the maximal capacity of about 0 29 bits per synapse in this study the author considers two possible sources for this discrepancy the first one is that the criteria for defining the maximal capacity are different with or without a constraint of perfect errorless storage the second one is a difference in the choice of the probability distribution of the random patterns used to compute this capacity this analysis shows in particular that for the willshaw model the maximal information capacity is much larger when the number of active neurons is exactly the same in every stored pattern than when it is given only in average in addition he gives an argument showing that this result may be generic e g valid for any activity level and independent of the learning rule

the conley index for fast slow systems ii multidimensional slow variable we develop a qualitative theory for fast slow systems with a one dimensional slow variable using conley index theory for singularity perturbed systems conditions are given which imply that if one can construct heteroclinic connections and periodic orbits in systems with the derivative of the slow variable set to 0 these orbits persist when the derivative of the slow variable is small and nonzero

error estimates for a finite element method for the drift diffusion semiconductor device equations the zero diffusion case in this paper new error estimates for an explicit finite element method for numerically solving the so called zero diffusion unipolar model a one dimensional simplified version of the drift diffusion semiconductor device equations are obtained the method studied in a previous paper combines a mixed finite element method using a continuous piecewise linear approxima tion of the electric field with an explicit upwinding finite element method using a piecewise constant approximation of the electron concentration by using a suitable extension of kuznetsov approximation theory for scalar conservation laws it is proved that under proper hypotheses on the data the l j error between the approximate and exact electron concentrations of the zero diffusion unipolar model is of order ax1 2 these estimates are sharp