the mckean vlasov equation in finite volume we study the mckean vlasov equation on the finite tori of length scale l in d dimensions we derive the necessary and sufficient conditions for the existence of a phase transition which are based on the criteria first uncovered in gates and penrose commun math phys 17 194 209 1970 and kirkwood and monroe j chem phys 9 514 526 1941 therein and in subsequent works one finds indications pointing to critical transitions at a particular model dependent value of the interaction parameter we show that the uniform density which may be interpreted as the liquid phase is dynamically stable for and prove abstractly that a critical transition must occur at however for this system we show that under generic conditions l large d 2 and isotropic interactions the phase transition is in fact discontinuous and occurs at some theta text t theta sharp finally for h stable bounded interactions with discontinuous transitions we show that with suitable scaling the theta text t l tend to a definitive non trivial limit as l

aggregation blowup and collapse the abc s of taxis in reinforced random walks in many biological systems movement of an organism occurs in response to a diffusible or otherwise transported signal and in its simplest form this can be modeled by diffusion equations with advection terms of the form first derived by patlak bull of math biophys 15 1953 pp 311 338 however other systems are more accurately modeled by random walkers that deposit a nondiffusible signal that modifies the local environment for succeeding passages in these systems one example of which is the myxobacteria the question arises as to whether aggregation is possible under suitable hypotheses on the transition rules and the production of a control species that modulates the transition rates davis probab theory related fields 84 1990 pp 203 229 has studied this question for a certain class of random walks and here we extend this analysis to the continuum limit of such walks we first derive several general classes of partial differential equations that depend on how the movement rules are

symplectic and multisymplectic lobatto methods for the good boussinesq equation in this paper we construct second order symplectic and multisymplectic integrators for the good boussineq equation using the two stage lobatto iiia iiib partitioned runge kutta method which yield an explicit scheme and is equivalent to the classical central difference approximation to the second order spatial derivative numerical dispersion properties and the stability of both integrators are investigated numerical results for different solitary wave solutions confirm the excellent long time behavior of symplectic and multisymplectic integrators by preserving local and global energy and momentum

a heuristic solution for fuzzy mixed model line balancing problem abstract this paper addresses the mixed model line balancing problem with fuzzy processing time a fuzzy binary linear programming model is formulated for the problem this fuzzy model is then transformed to a mixed zero one program due to the complexity nature in handling fuzzy computation new approximated fuzzy arithmetic operation is presented a fuzzy heuristic is developed to solve this problem based on the aggregating fuzzy numbers and combined precedence constraints the general idea of our approach is to arrange the jobs in a sequence by a varying section exchange procedure then jobs are allocated into workstations based on these aggregated fuzzy times with the considerations of technological constraint and cycle time limit promising results are obtained by experiments

reinforced random walk leta i i 1 be a sequence of nonnegative numbers difine a nearest neighbor random motion overrightarrow x x 0 x 1 on the integers as follows initially the weight of each interval i i 1 i an integer equals 1 if at timen an interval i i 1 has been crossed exactlyk times by the motion its weight is 1 sum limits j 1 k a j given x 0 x 1 x n i0 i1 in the probability thatx n 1 isi n 1 ori n 1 is proportional to the weights at timen of the intervals i n 1 i n and i n iin 1 we prove that overrightarrow x either visits all integers infinitely often a s or visits a finite number of integers eventually oscillating between two adjacent integers a s and that mathop lim limits n to infty x n n 0 a s for much more general reinforcement schemes we provep overrightarrow x visits all integers infinitely often p overrightarrow x has finite range 1

nonlinear stability and convergence of finite difference methods for the good boussinesq equation the good boussinesq equationu tt u xxxx u xx u 2 xx has recently been found to possess an interesting soliton interaction mechanism in this paper we study the nonlinear stability and the convergence of some simple finite difference schemes for the numerical solution of problems involving the good boussinesq equation numerical experimentas are also reported

on a singular perturbation problem arising in the theory of evolutionary distributions evolution by natural selection is a process by which progeny inherit some properties from their progenitors with small variation these properties are subject to natural selection and are called adaptive traits and carriers of the latter are called phenotypes the distribution of the density of phenotypes in a population is called evolutionary distributions ed we analyze mathematical models of the dynamics of a system of ed such systems are anisotropic in that diffusion of phenotypes in each population equation remains positive in the directions of their own adaptive space and vanishes in the directions of the other s adaptive space we prove that solutions to such systems exist in a sense weaker than the usual we develop an algorithm for numerical solutions of such systems finally we conduct numerical experiments with a model in which populations compete that allow us to observe salient attributes of a specific system of ed

a system of reaction diffusion equations arising in the theory of reinforced random walks we investigate the properties of solutions of a system of chemotaxis equations arising in the theory of reinforced random walks we show that under some circumstances finite time blow up of solutions is possible in other circumstances the solutions will decay to a spatially constant solution collapse we also give some intuitive arguments which demonstrate the possibility of the existence of aggregation piecewise constant solutions

numerical simulation of interaction between schrodinger field and klein gordon field by multisymplectic method abstract we propose a multisymplectic scheme to solve the coupled klein gordon schrodinger system the scheme preserves the multisymplectic geometry structure exactly by satisfying the discrete multisymplectic conservation law and can simulate the original waves well in a long time this scheme also has discrete quasi norm conservation law numerical experiments demonstrate the consistency between the theoretical analysis and the numerical results

multisymplectic geometry local conservation laws and fourier pseudospectral discretization for the good boussinesq equation the multisymplectic geometry for the good boussinesq equation is presented in this paper the multisymplectic form and the local energy and momentum conservation laws are derived directly from the variational principle the multisymplectic hamiltonian formulation is also presented based on the multisymplectic formulation we investigate the corresponding multisymplectic fourier pseudospectral discretization

multisymplectic geometry variational integrators and nonlinear pdes this paper presents a geometric variational approach to continuous and discrete mechanics and field theories using multisymplectic geometry we show that the existence of the fundamental geometric structures as well as their preservation along solutions can be obtained directly from the variational principle in particular we prove that a unique multisymplectic structure is obtained by taking the derivative of an action function and use this structure to prove covariant generalizations of conservation of symplecticity and noether s theorem natural discretization schemes for pdes which have these important preservation properties then follow by choosing a discrete action functional in the case of mechanics we recover the variational symplectic integrators of veselov type while for pdes we obtain covariant spacetime integrators which conserve the corresponding discrete multisymplectic form as well as the discrete momentum mappings corresponding to symmetries we show that the usual notion of symplecticity along an infinite dimensional space of fields can be naturally obtained by making a spacetime split all of the aspects of our method are demonstrated with a nonlinear sine gordon equation including computational results and a comparison with other discretization schemes

soliton and antisoliton interactions in the good boussinesq equation the solitary wave interaction mechanism for the good boussinesq equation is investigated and found to be far more complicated than was previously thought three salient features are that solitary waves only exist for a finite range of velocities that large solitons can turn into so called antisolitons and that it is possible for solitons to merge and split small solitons however appear to be stable the existence of a potential well is linked to the different behaviors observed between small and large initial conditions

branch and bound algorithms for the multi product assembly line balancing problem abstract this paper considers a flexible manufacturing system for several products each requiring a number of sequential nonpreemptive tasks some of which may be common to many products work stations are capable of handling any task the objective is to minimize the number of work stations necessary to manufacture all products so that tasks are performed in the proper order and so that all tasks assigned to any given station can be executed within a prescribed time frame a new branch and bound approach for this problem is presented it contains two main improvements over previously published tree search algorithms an improved lower bounding procedure and a more powerful partitioning scheme the algorithm can be used either as a heuristic in its truncated version or as an exact method in its full version tests performed on randomly generated problems confirm the effectiveness of the proposed improvements

on a nonlocal phase separation model abstract an alternative to the cahn hilliard model of phase separation for two phase systems in a simplified isothermal case is given the model is derived from a free energy with a nonlocal interacting term and allows reasonable bounds for the concentrations using the free energy as lyapunov functional the asymptotic state of the system is investigated and characterized by a variational principle

many body theory of chemotactic cell cell interactions we consider an individual based stochastic model of cell movement mediated by chemical signaling fields this model is formulated using langevin dynamics which allows an analytic study using methods from statistical and many body physics in particular we construct a diagrammatic framework within which to study cell cell interactions in the mean field limit where statistical correlations between cells are neglected we recover the deterministic keller segel equations within exact perturbation theory in the chemotactic coupling epsilon statistical correlations are non negligible at large times and lead to a renormalization of the cell diffusion coefficient d r an effect that is absent at mean field level an alternative closure scheme based on the necklace approximation probes the strong coupling behavior of the system and predicts that d r is renormalized to zero at a critical value of epsilon indicating self localization of the cell stochastic simulations of the model give very satisfactory agreement with the perturbative result at higher values of the coupling simulations indicate that d r approximately epsilon 2 a result at odds with the necklace approximation we briefly discuss an extension of our model which incorporates the effects of short range interactions such as cell cell adhesion

on multisymplecticity of partitioned runge kutta methods previously it has been shown that discretizing a multi hamiltonian pde in space and time with partitioned runge kutta methods gives rise to a system of equations that formally satisfy a discrete multisymplectic conservation law however these previous studies use the same partitioning of the variables into two parts in both space and time this gives rise to a large number of cases to be considered each with its own set of conditions to be satisfied we present here a much simpler set of conditions covering all of these cases where the variables are partitioned independently in space and time into an arbitrary number of parts in general it is not known when such a discretization of a multi hamiltonian pde will give rise to a well defined numerical integrator however a numerical integrator that is explicit will typically be well defined in this paper we give sufficient conditions on a multi hamiltonian pde for a lobatto iiia iiib discretization in space to give rise to explicit odes and an algorithm for constructing these odes

on explosions of solutions to a system of partial differential equations modelling chemotaxis a system of partial differential equations modelling chemotactic aggregation is analysed keller segel model conditions on the system of paramaters are given implying global existence of smooth solutions in two space dimensions and radially symmetric situations explosion of the bacteria concentration in finite time is shown for a class of initial values

stationary solutions of chemotaxis systems the keller segel model is a system of partial differential equations modelling a mutual attraction of amoebae caused by releasing a chemical substance chemotaxis this paper analyzes the stationary solutions of the system with general nonlinearities via bifurcation techniques and gives a criterion for bifurcation of stable nonhomogeneous aggregation patterns examples are discussed with various kinds of nonlinearities modelling the sensitivity of the chemotaxis response

on the derivation of macroscopic hyperbolic equations for binary multicellular growing mixtures this paper presents an asymptotic theory based on the hyperbolic scaling for a large class of boltzmann type equations suitable to model the evolution of a binary mixture of multicellular systems in biology the modelling approach is that of the kinetic theory for active particles time scaling related to cell movement and biological activity are shown to play a crucial role in determining the macroscopic equations corresponding to each case applications are focused on the immune competition

from a discrete model of chemotaxis with volume filling to a generalized patlak keller segel model we present a discrete model of chemotaxis whereby cells responding to a chemoattractant are seen as individual agents whose movement is described through a set of rules that result in a biased rand

a geometric formulation of the conservation of wave action and its implications for signature and the classification of instabilities action symplecticity signature and complex instability are fundamental concepts in hamiltonian dynamics which can be characterized in terms of the symplectic structure in this paper hamiltonian pdes on unbounded domains are characterized in terms of a multisymplectic structure where a distinct differential two form is assigned to each space direction and time this leads to a new geometric formulation of the conservation of wave action for linear and nonlinear hamiltonian pdes and via stokes s theorem a conservation law for symplecticity each symplectic structure is used to define a signature invariant on the eigenspace of a normal mode the first invariant in this family is classical krein signature or energy sign when the energy is time independent and the other spatial signatures are energy flux signs leading to a classification of instabilities that includes information about directional spatial spreading of an instability the theory is applied to several examples the boussinesq equation the water wave equations linearized about an arbitrary stokes s wave rotating shallow water flow and flow past a compliant surface some implications for non conservative systems are also discussed

asymptotic stability of a two species chemotaxis system with non diffusive chemoattractant abstract we study the behavior of two biological populations u and v attracted by the same chemical substance whose behavior is described in terms of second order parabolic equations the model considers a logistic growth of the species and the interactions between them are relegated to the chemoattractant production the system is completed with a third equation modeling the evolution of chemical we assume that the chemical w is a non diffusive substance and satisfies an ode more precisely u t u u 1 w w 1 u 1 u x t 0 v t v v 2 w w 2 v 1 v x t 0 w t h u v w x t 0 under appropriate boundary and initial conditions in an n dimensional open and bounded domain we consider the cases of positive chemo sensitivities not necessarily constant elements the chemical production function h increases as the concentration of the species u and v increases we first study the global existence and uniform boundedness of the solutions by using an iterative approach the asymptotic stability of the homogeneous steady state is a consequence of the growth of h i and the size of i finally some examples of the theoretical results are presented for particular functions h and i

the derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many particle systems the chemotaxis equations are a well known system of partial differential equations describing aggregation phenomena in biology in this paper they are rigorously derived from an interacting stochastic many particle system where the interaction between the particles is rescaled in a moderate way as population size tends to infinity the novelty of this result is that in all previous applications of this kind of limiting procedure the principal part of the system is assumed to fulfill an ellipticity condition which is not satisfied in our case new techniques which deal with this difficulty are presented

stability of solutions of chemotaxis equations in reinforced random walks in this paper we consider a nonlinear system of differential equations consisting of one parabolic equation and one ordinary differential equation the system arises in chemotaxis a process whereby living organisms respond to chemical substance by moving toward higher or lower concentrations of the chemical substance or by aggregating or dispersing we prove that stationary solutions of the system are asymptotically stable

derivation of lagrangian density for the good boussinesq equation and multisymplectic disretizations abstract the lagrangian density for the good boussinesq equation is derived by introducing a new variable in this paper based on this density we consider the corresponding multisymplectic formulation and discretizations numerical experiments are also reported

the diffusion limit of transport equations derived from velocity jump processes in this paper we study the diffusion approximation to a transport equation that describes the motion of individuals whose velocity changes are governed by a poisson process we show that under an appropriate scaling of space and time the asymptotic behavior of solutions of such equations can be approximated by the solution of a diffusion equation obtained via a regular perturbation expansion in general the resulting diffusion tensor is anisotropic and we give necessary and sufficient conditions under which it is isotropic we also give a method to construct approximations of arbitrary high order for large times in a second paper part ii we use this approach to systematically derive the limiting equation under a variety of external biases imposed on the motion depending on the strength of the bias it may lead to an anisotropic diffusion equation to a drift term in the flux or to both our analysis generalizes and simplifies previous derivations that lead to the classical patlak keller segel alt

existence of solutions of the hyperbolic keller segel model we are concerned with the hyperbolic keller segel model with quorum sensing a model describing the collective cell movement due to chemical signalling with a flux limitation for high cell densities this is a first order quasilinear equation its flux depends on space and time via the solution to an elliptic pde in which the right hand side is the solution to the hyperbolic equation this model lacks strong compactness or contraction properties our purpose is to prove the existence of an entropy solution obtained as usual in passing to the limit in a sequence of solutions to the parabolic approximation the method consists in the derivation of a kinetic formulation for the weak limit the specific structure of the limiting kinetic equation allows for a rigidity theorem which identifies some property of the solution which might be nonunique to this kinetic equation this is enough to deduce a posteriori the strong convergence of a subsequence

infinite time aggregation for the critical patlak keller segel model in ℝ2 we analyze the two dimensional parabolic elliptic patlak keller segel model in the whole euclidean space r2 under the hypotheses of integrable initial data with finite second moment and entropy we first show local in time existence for any mass of free energy solutions namely weak solutions with some free energy estimates we also prove that the solution exists as long as the entropy is controlled from above the main result of the paper is to show the global existence of free energy solutions with initial data as before for the critical mass 8 actually we prove that solutions blow up as a delta dirac at the center of mass when t keeping constant their second moment at any time furthermore all moments larger than 2 blow up as t if initially bounded

multi symplectic spectral discretizations for the zakharov kuznetsov and shallow water equations abstract the time evolution of multi symplectic pdes with periodic boundary conditions in one and two space dimensions is considered we introduce the idea of a multi symplectic fourier transform for systems on periodic domains leading to a semi discretization on fourier space and to the concept of multi symplecticity on fourier space the spatial discretization leads to a discrete lattice model with certain discrete conservation laws imposed on it a discrete wave model a by product of the semi discretization is that it leads automatically to a system of hamiltonian odes in time when truncated we show that the one dimensional shallow water equations and the two dimensional zakharov kuznetsov equation are multi symplectic and derive spectral discretizations for these systems and present numerical experiments

from microscopic to macroscopic description of multicellular systems and biological growing tissues this paper presents an asymptotic theory for a large class of boltzmann type equations suitable to model the evolution of multicellular systems in biology the mathematical approach described herein shows how various types of diffusion phenomena linear and nonlinear can be obtained in suitable asymptotic limits time scaling related to cell movement and biological activity are shown to play a crucial role in determining the macroscopic equations corresponding to each case

the mixed and multi model line balancing problem a comparison abstract the balancing problem deals with the assignment of tasks to work stations we can distinguish two approaches in the literature on the mixed model line balancing problem that both transform this problem into a single model line balancing problem these approaches use combined precedence diagrams and adjusted task processing times respectively an experiment was carried out to compare several heuristics based on the combined precedence diagram a new optimisation method has been developed the results indicate that the position of common tasks in the precedence diagram of the different models has a significant effect on both the cpu time and the unequal distribution of the total work content of single models among work stations moreover good solutions with respect to the number of required stations go together with long cpu times for several instances we decreased the cpu times considerably without deteriorating the performance of the methods by using a reversed combined precedence diagram

singularity patterns in a chemotaxis model the authors study a chemotactic model under certain assumptions and obtain the existence of a class of solutions which blow up at the center of an open disc in finite time such a finite time blow up of solutions implies chemotactic collapse namely concentration of species to form sporae the model studied is the limiting case of a basic chemotactic model when diffusion of the chemical approaches infinity which has the form ut u u v 0 v u 1 on r2 where is an open disc with no flux homogeneous neumann boundary conditions the initial conditions are continuous functions u x 0 u0 x 0 v x 0 v0 x 0 for x under these conditions the authors prove there exists a radially symmetric solution u r t which blows up at r 0 t t a specific description of such a solution is presented the authors also discuss the strong similarity between the chemotactic model they study and the classical stefan problem

uniform convergence of proliferating particles to the fkpp equation abstract in this paper we consider a system of brownian particles with proliferation whose rate depends on the empirical measure the dependence is more local than a mean field one and has been called moderate interaction by oelschlager 16 17 we prove that the empirical process converges uniformly in the space variable to the solution of the fisher kolmogorov petrowskii piskunov equation we use a semigroup approach which is new in the framework of these systems and is inspired by some literature on stochastic partial differential equations

pseudospectral method for the good boussinesq equation we prove the nonlinear stability and convergence of a fully discrete pseudospectral scheme for the good boussinesq equation utt uxxxx uxx u2 xx numerical comparisons with finite difference schemes are also reported

on a parabolic elliptic system with chemotaxis and logistic type growth we consider a nonlinear pdes system of two equations of parabolic elliptic type with chemotactic terms the system models the movement of a biological population u towards a higher concentration of a chemical agent w in a bounded and regular domain rn for arbitrary n n after normalization the system is as follows ut u div um w u 1 u in t 0 t w w u in t for some positive constants m and with positive initial datum u0 and neumann boundary conditions we study the range of parameters and constrains for which the solution exists globally in time if either m 1 or m 1 and n 22 m 1 n the solution exists globally in time moreover if m 1 and 2 and there exist positive constants u 0 and u 0 such that 0 u 0 u0 u 0 we have that u 1 l w 1 l 0 as t