scaling of reaction fronts under quenched disorder abstract the dynamics of fronts in diffusion limited reactions a b c with initially segregated reactants in strongly disordered media is studied the scaling of the front width w in the asymptotic limit w t is analyzed as a function of the disorder strength and spatial dimensionality d we show that exponent a in d dimensions is a monotonic decreasing function of disorder strength as measured from diffusive properties of reactants we also find that a has a mean field behavior for d 2 scaling anomalies are observed in related quantities such as nearest neighbor distance midpoint fluctuations and concentration profiles of reactants near the center of the front

reaction kinetics of a b 0 a method to find the exact reaction rate density function of an a b 0 reaction on a finite 1d bar is derived we consider the case where the a and b species diffuse at the same rate have equal initial numbers with the initial location of either species on the bar being arbitrary the extension of the method to study a b 0 on fractals is also shown additionally we introduce the notion of a continual in time reaction front and a ceasing in time reaction front and provide a physical criterion for when either occurs

markov chain of distances between parked cars we describe the distribution of distances between parked cars as a solution of certain markov processes and show that its solution is obtained with the help of a distributional fixed point equation under certain conditions the process is solved explicitly the resulting probability density is compared with the actual parking data measured in the city

applications of field theoretic renormalization group methods to reaction diffusion problems we review the application of field theoretic renormalization group rg methods to the study of fluctuations in reaction diffusion problems we first investigate the physical origin of universality in these systems before comparing rg methods to other available analytic techniques including exact solutions and smoluchowski type approximations starting from the microscopic reaction diffusion master equation we then pedagogically detail the mapping to a field theory for the single species reaction ka a k we employ this particularly simple but non trivial system to introduce the field theoretic rg tools including the diagrammatic perturbation expansion renormalization and callan symanzik rg flow equation we demonstrate how these techniques permit the calculation of universal quantities such as density decay exponents and amplitudes via perturbative dc d expansions with respect to the upper critical dimension dc with these basics established we then provide an overview of more sophisticated applications to multiple species reactions disorder effects l evy flights persistence problems and the influence of spatial boundaries we also analyse field theoretic approaches to non equilibrium phase transitions separating active from absorbing states we focus particularly on the generic directed percolation universality class as well as on the most prominent exception to this class even offspring branching and annihilating random walks finally we summarize the state of the field and present our perspective on outstanding problems for the future

time dependent ginzburg landau equations of superconductivity abstract we study in this article the existence uniqueness and long time behavior of the solutions of a nonstationary ginzburg landau superconductivity model we first prove the existence and uniqueness of solutions with h 1 initial data which are crucial for the study of the global attractor we also obtain for the first time the existence of global weak solutions of the model with l 2 initial data it is then proved that the ginzburg landau system admits a global attractor which represents exactly all the long time dynamics of the system the global attractor obtained consists of exactly the set of steady state solutions and its unstable manifold its hausdorff and fractal dimensions are estimated in terms of the physically relevant ginzburg landau parameter diffusion parameter and applied magnetic field we construct explicitly absorbing sets for some abstract semigroups having a lyapunov functional and consequently prove the existence of global attractors this abstract result is applied to the ginzburg landau system for which the existence of global attractor does not seem to be the direct consequence of some a priori estimates of solutions

symmetric vortices for the ginzberg landau equations of superconductivity and the nonlinear desingularization phenomenon abstract the existence of countably many distinct symmetric vortices for the nonlinear ginzberg landau equations of superconductivity with an arbitrary positive parameter is proven we then prove the existence of a linearization phenomenon as the parameter in this case the linearized equations contain an appropriate dirac delta function these equations are known as the london equations

approximations of a ginzburg landau model for superconducting hollow spheres based on spherical centroidal voronoi tessellations in this paper the numerical approximations of the ginzburg landau model for a superconducting hollow spheres are constructed using a gauge invariant discretization on spherical centroidal voronoi tessellations a reduced model equation is used on the surface of the sphere which is valid in the thin spherical shell limit we present the numerical algorithms and their theoretical convergence as well as interesting numerical results on the vortex configurations properties of the spherical centroidal voronoi tessellations are also utilized to provide a high resolution scheme for computing the supercurrent and the induced magnetic field

convergence analysis of a numerical method for a mean field model of superconducting vortices in the mean field models of superconductivity the individual vortex like structures occurring in practical type ii superconductors are averaged and a vortex density is solved for them the numerical solution of the mean field models makes large scale simulations of vortex phenomena possible in this paper we present a simple convergence analysis for a numerical method based on hybrid finite element finite volume finite difference approximations in the two dimensional case by providing various different interpretations to the discretization scheme

inhomogeneous two species annihilation in the steady state the authors investigate steady state geometrical properties of a reaction interface in the two species annihilation process a b to 0 when a flux j of a and b particles is injected at opposite extremities of a finite domain by balancing the input flux with the number of reactions they determine that the width w of the reaction zone scales as j 1 3 in the large flux limit and that the concentration in this zone is proportional to j2 3 this same behaviour is deduced from the solution to the reaction diffusion equation in the low flux limit the concentration is almost independent of position and is proportional to j in the latter case the local reaction rate reaches maximum at the edges of the system rather than at the midpoint when the two species approach at a finite velocity there exists a critical velocity above which the reactants essentially pass through each other results similar to those in one dimension are found in two and three dimensional radial geometries finally they apply the quasistatic approximation to their steady state solution to recover the known time dependence for the reaction zone width for the case of initially separated components with no external input

geometrical aspects of stability theory for hill s equations where a and b are real parameters apparently the first stability diagram was drawn in the classical paper by b van der pol m j o sti utt 15 since then many papers and textbooks have appeared on the subject we mention j j stoker 13 j meixner f w sch fke 12 d m levy 8 j b keller 11 m i weinstein j b keller 16 17 j poschel e trubowltz 9 and v i arnold 3 5 6 this literature contains extensive estimates on the order of tangency of resonance tongues and the behavior of stability boundaries at infinity a full understanding of the stability diagrams is still lacking for instance the appearance of instability pockets in some cases see figure 1 cf 15 calls for a geometric explanation it should be noted that this phenomenon does not occur in the classical mathieu case cf 12 for a preliminary study near resonances using singularity theory we refer to afsharnejad 1 related bifurcational aspects of nonlinear perturbations of mathieu s equation near resonances were studied by broer vegter 7 the global geometry of the symplectic group was used for a stability proof of a hill equation in levi 10 we introduce the main object of study of this paper

a reversible bifurcation analysis of the inverted pendulum the inverted pendulum with a periodic parametric forcing is considered as a bifurcation problem in the reversible setting parameters are given by the size of the forcing and the frequency ratio normal form theory provides an integrable approximation of the poincare map generated by a planar vector field genericity of the model is studied by a perturbation analysis where the spatial symmetry is optional here equivariant singularity theory is used

a crank nicolson finite element method and the optimal error estimates for the modified time dependent maxwell schrodinger equations in this paper we study a crank nicolson finite element method for the modified maxwell schrodinger equations which are deduced from the time dependent maxwell schrodinger equations under some assumptions we present an almost unconditionally optimal error estimate for the numerical algorithm which only requires the time step delta t 1 the key to our analysis is twofold for one thing we use some tricks to tackle the tough nonlinear terms without applying the inverse inequality to estimate the l infty norm of the numerical solutions and get rid of certain restrictions like delta t leq c h alpha on the time step for another we employ a new technique to weaken the time step restriction required by the application of the discrete gronwall inequality numerical tests are carried out to verify the theoretical results

spatial organization in two species annihilation the spatial structure of reactants in the two species annihilation reaction a b 0 is described in one dimension we investigate the distribution of domain sizes and the distributions of nearest neighbor distances between particles of the same and of opposite species the latter two quantities are characterized by a new length scale which is intermediate to the domain size t1 2 and the typical interparticle spacing t1 4 a scaling argument suggests that the typical distance between particles of opposite species or equivalently the gaps between domains grows ast with 3 8 and 1 3 respectively in spatial dimensiond 1 and 2 the average density profile of a single domain is spatially nonuniform with the density decaying to zero linearly as the domain edge is approached this behavior permits a determination of the distribution of nearest neighbor distances of same species reactants the corresponding moments of this distribution exhibit multiscaling which involves geometric averages of different powers of the domain size and the interparticle spacing

adaptive galerkin methods with error control for a dynamical ginzburg landau model in superconductivity the time dependent ginzburg landau model which describes the phase transitions taking place in superconductors is a coupled system of nonlinear parabolic equations it is discretized semi implicitly in time and in space via continuous piecewise linear finite elements a posteriori error estimates are derived for the l infty l 2 norm by studying a dual problem of the linearization of the original system other than the dual of error equations numerical simulations are included which illustrate the reliability of the estimators and the flexibility of the proposed adaptive method

nearest neighbour distances of diffusing particles from a single trap we present a simple method for obtaining the typical smallest distance from an absorbing trap to the nearest particle in a system of brownian particles in terms of the overall density profile we also obtain the distribution function for the minimum distance by using a quasistatic approximation for the diffusion equation we derive new results for two dimensions the characteristic distance to the nearest particle increases asymptotically as m the technique is useful in diffusion reaction systems in the smoluchowski model for bimolecular reactions an ideal spherical trap centred at the origin is surrounded by a cloud of brownian particles which are captured upon contact with the trap the reaction rate in this formulation depends on the spatial distribution of particles about the trap recently interest has focused on finer details of this distribution such as the distance of the nearest surviving particle from the trap l 31 the density distribution function of that distance is the key to several exact results concerning some diffusion reaction systems in one dimension 4 while the spatial density of the particles surrounding the trap can be easily obtained by solving a diffusion equation with the appropriate boundary conditions the distance to the nearest particle from the trap seems to require a more sophisticated analysis 13 in this paper we suggest a simple derivation for the typical distance from the trap to the nearest particle and of the distribution of those distances based upon the more readily attainable spatial density of all particles and simple facts from the statistics of extremes 5 6 to demonstrate our method we rederive the known results for one and three dimensions the simplicity of our approach together with using a quasistatic approximation 7 for solving the diffusion equation allows us to solve the two dimensional case for the first time consider first the one dimensional problem a perfect trap is located at the origin and is initially surrounded by a homogeneous density co of brownian particles which diffuse with a diffusion constant d the density c x t of the surviving particles to the right of the trap is given by the diffusion equation

finite element approximation of the cahn hilliard equation on surfaces abstract in this paper we consider the phase separation on general surfaces by solving the nonlinear cahn hilliard equation using a finite element method a fully discrete approximation scheme is introduced and we establish a priori estimates for the discrete solution that does not rely on any knowledge of the exact solution beyond the initial time this in turn leads to convergence and optimal error estimates of the discretization scheme numerical examples are also provided to substantiate the theoretical results

reaction diffusion branching models of stock price fluctuations abstract several models of stock trading bak et al physica a 246 1997 430 are analyzed in analogy with one dimensional two species reaction diffusion branching processes using heuristic and scaling arguments we show that the short time market price variation is subdiffusive with a hurst exponent h 1 4 biased diffusion towards the market price and blind eyed copying lead to crossovers to the empirically observed random walk behavior h 1 2 at long times the calculated crossover forms and diffusion constants are shown to agree well with simulation data

modelling gap size distribution of parked cars using random matrix theory abstract we apply the random matrix theory to the car parking problem for this purpose we adopt a coulomb gas model that associates the coordinates of the gas particles with the eigenvalues of a random matrix the nature of interaction between the particles is consistent with the tendency of the drivers to park their cars near to each other and in the same time keep a distance sufficient for manoeuvring we show that the recently measured gap size distribution of parked cars in a number of roads in central london is well represented by the spacing distribution of a gaussian unitary ensemble

stability analysis for systems of nonlinear hill s equations systems of nonlinear dierential equations with periodic coecients which include hill s and mathieu s equations as examples in the linear limit are important from a practical point of view nonlinear hill s equations model a variety of dynamical systems of interest to physics and engineering in which perturbations enter as periodic modulations of their linear frequencies as is well known the stability properties of some fundamental periodic solutions of these systems is often an essential problem the main purpose of this paper is to concentrate on one such class of nonlinear hill s equations and study the stability properties of some of their simplest periodic solutions analytically as well as numerically to accomplish this task we rst use an extension of the generalized averaging method to approximate these solutions and then apply the technique of multiple scaling to perform the stability analysis a three particle system with free free boundary conditions is studied as an example the accuracy of our results is tested within the limits of rst order perturbation theory and is found to be well conrmed by numerical experiments the stability analysis of these simple periodic solutions though local in itself can yield considerable information about more global properties of the dynamics since it is in the vicinity of such solutions that the largest regions of regular or chaotic motion are usually observed depending on whether the periodic solution is respectively stable or unstable c 2000 elsevier science b v all rights reserved

diffusion controlled reaction a b c with initially separated reactants we review recent developments in the study of diffusion reaction systems of the type a b c in which the reactants are initially separated we consider two initial boundary conditions i the a and b particles are initially placed uniformly in euclidean space at x 0 and x 0 respectively and ii the a particles are diffusing and inserted at a given site and the b particles are static and distributed uniformly in space we present analytical and numerical results for both systems we consider d 1 2 3 dimensional systems as well as fractal lattices

analysis and convergence of the mac scheme ii navier stokes equations the mac discretization scheme for the incompressible navier stokes equations is interpreted as a covolume approximation to the equations using some results from earlier papers dealing with covolume error estimates for div curl equation systems and under certain conditions on the data and the solutions of the navier stokes equations we obtain first order error estimates for both the vorticity and the pressure

exact results in trapping reactions for mobile particles and a single trap we have exploited a stochastic model for the description of diffusion controlled reactions for trapping processes with a single trap in a one dimensional lattice in order to describe in an exact way the particle distribution as seen from the trap being it fix or mobile as well as for perfect or imperfect reactions from this we obtain the exact result for the total number of absorbed particles moreover we get a formal expression for the nearest neighbor particle distribution that gives the known results in the limit cases of fixed trap and fixed particles

constrained centroidal voronoi tessellations for surfaces centroidal voronoi tessellations are useful for subdividing a region in euclidean space into voronoi regions whose generators are also the centers of mass with respect to a prescribed density function of the regions their extensions to general spaces and sets are also available for example tessellations of surfaces in a euclidean space may be considered in this paper a precise definition of such constrained centroidal voronoi tessellations ccvts is given and a number of their properties are derived including their characterization as minimizers of an energy deterministic and probabilistic algorithms for the construction of ccvts are presented and some analytical results for one of the algorithms are given computational examples are provided which serve to illustrate the high quality of ccvt point sets finally ccvt point sets are applied to polynomial interpolation and numerical integration on the sphere

diffusion controlled bimolecular reactions long and intermediate time regimes with imperfect trapping within a galanin approach we study a model for the a b ensuremath rightarrow b diffusion reaction system in one dimension in which the reaction sites are modeled by a markovian stochastic process mean values of the a particle density are obtained exactly for dilute b systems and approximately for dense ones results are compared with numerical simulations and show good agreement in the short intermediate and long time regimes

stability regions for coupled hill s equations abstract in this paper we extend well known results for one hill s equation and present the stability analysis of two coupled hill s equations for which the general theory is not readily available approximate expressions are derived in the context of peturbation theory for the boundaries between bounded and unbounded periodic solutions with frequencies n m n and m are positive integers of both linear and nonlinear coupled mathieu equations as examples excellent agreement is found between theoretical predictions and numerical computations over large ranges of parameter values and initial conditions these periodic solutions are important because they correspond to some of the lowest order resonances of the system and when they are stable they turn out to have large regions of regular motion around them in phase space coupled mathieu equations appear in numerous important physical applications in problems of accelerator dynamics electrohydrodynamics and mechanics

analysis and approximation of the ginzburg landau model of superconductivity the authors consider the ginzburg landau model for superconductivity first some well known features of superconducting materials are reviewed and then various results concerning the model the resultant differential equations and their solution on bounded domains are derived then finite element approximations of the solutions of the ginzburg landau equations are considered and error estimates of optimal order are derived

analysis and convergence of the mac scheme i the linear problem the mac marker and cell discretization of fluid flow is analysed for the stationary stokes equations it is proved that the discrete approximations do in fact converge to the exact solutions of the flow equations estimates using mesh dependent norms analogous to the standard bf h 1 and l 2 norms are given for the velocity and pressure respectively

modelling the gap size distribution of parked cars abstract we have measured the distribution of distances between parked cars in a number of roads in central london we compare the results with models of random sequential adsorption or random car parking models as they are often called our empirical results do not agree with these models and hence we introduce alternative models for the parking process the first model we propose is one where cars can only park if the space is greater than or equal to a chosen minimum size where the extra space can be used for manoeuvering the results of this model suggest that in addition to a deposition process a re positioning process is required to explain our empirical results hence we introduce a further model where at each time step with probability p a car is randomly placed on a line if the space is empty and with probabilty 1 p is placed in a space but then rolls to the nearest parked car so that it is y distance from the nearest car with probability f y we concentrate on two specific probabilities f y and find that for one case the gap size distribution agrees with the empirical results

multiparticle trapping problem in the half line abstract a variation of rosenstock s trapping model in which n independent random walkers are all initially placed upon a site of a one dimensional lattice in the presence of a one sided random distribution with probability c of absorbing traps is investigated the probability survival probability n t that no random walker is trapped by time t for n 1 is calculated by using the extended rosenstock approximation this requires the evaluation of the moments of the number sn t of distinct sites visited in a given direction up to time t by n independent random walkers the rosenstock approximation improves when n increases working well in the range dt ln 2 1 c ln n d being the diffusion constant the moments of the time lifetime before any trapping event occurs are calculated asymptotically too the agreement with numerical results is excellent

transition curves for the quasi periodic mathieu equation in this work we investigate an extension of mathieu s equation the quasi periodic qp mathieu equation given by ddot psi delta eps cos t cos omega t psi 0 for small eps and irrational omega of interest is the generation of stability diagrams that identify the points or regions in the delta omega parameter plane for fixed eps for which all solutions of the qp mathieu equation are bounded numerical integration is employed to produce approximations to the true stability diagrams both directly and through contour plots oflyapunov exponents in addition we derive approximate analytic expressions for transition curves using two distinct techniques 1 a regular perturbation method under which transition curves delta delta omega eps are each expanded in powers of eps and 2 the method of harmonic balance utilizing hill s determinants both analytic methods deliver results in good agreement with those generated numerically in the sense that pre

sweeping algorithms for inverting the discrete ginzburg landau operator the ginzburg landau equations we study arise in the modeling of superconductivity one widely used method of discretizing the equations together with the associated periodic boundary conditions in the case of a rectangle of dimension 2 leads to a five point stencil solving the system means inverting a sparse matrix of dimension n 2 where n is the number of grid points on each side of the rectangle we propose a method that is similar to the shooting technique in the numerical solution of ordinary differential equations for small n the method requires inverting a full matrix of dimension 2n when n is large an iterative procedure combining partial sweeping and the technique of divide and conquer domain decomposition is appropriate

spatial structure in diffusion limited two particle reactions we analyze the limiting behavior of the densities a t and b t and the random spatial structure r a t b t for the diffusion controlled chemical reaction a b inert for equal initial densities b 0 b 0 there is a change in behavior fromd 4 where a t b t c td 4 tod 4 where a t b t c t ast the termc depends on the initial densities and changes withd there is a corresponding change in the spatial structure ind 4 both particle types are after large times present locally in concentrations not depending on type or location ind 4 both particle types are present locally but with random concentrations and the process tends to a limit

on the size distribution of poisson voronoi cells abstract poisson voronoi diagrams are useful for modeling and describing various natural patterns and for generating random lattices although this particular space tessellation is intensively studied by mathematicians in two and three dimensional 3d spaces there is no exact result known for the size distribution of voronoi cells motivated by the simple form of the distribution function in the 1d case a simple and compact analytical formula is proposed for approximating the voronoi cell s size distribution function in the practically important 2d and 3d cases as well denoting the dimensionality of the space by d d 1 2 3 the f y const y 3 d 1 2 exp 3 d 1 y 2 compact form is suggested for the normalized cell size distribution function by using large scale computer simulations the viability of the proposed distribution function is studied and critically discussed

voronoi cell patterns theoretical model and applications we use a simple fragmentation model to describe the statistical behavior of the voronoi cell patterns generated by a homogeneous and isotropic set of points in 1d and in 2d in particular we are interested in the distribution of sizes of these voronoi cells our model is completely defined by two probability distributions in 1d and again in 2d the probability to add a new point inside an existing cell and the probability that this new point is at a particular position relative to the preexisting point inside this cell in 1d the first distribution depends on a single parameter while the second distribution is defined through a fragmentation kernel in 2d both distributions depend on a single parameter the fragmentation kernel and the control parameters are closely related to the physical properties of the specific system under study we use our model to describe the voronoi cell patterns of several systems specifically we study the island nucleation with irreversible attachment the 1d car parking problem the formation of second level administrative divisions and the pattern formed by the paris metro stations

direct discretization of planar div curl problems a control volume method is proposed for planar div curl systems the method is independent of potential and least squares formulations and works directly with the div curl system the novelty of the technique lies in its use of a single local vector field component and two control volumes rather than the other way round a discrete vector field theory comes quite naturally from this idea and is developed in the paper error estimates are proved for the method and other ramifications investigated

fokker planck description for a linear delayed langevin equation with additive gaussian noise we construct an equivalent probability description of linear multi delay langevin equations subject to additive gaussian white noise by exploiting the time convolutionless transform and a time variable transformation we are able to write a fokker planck equation fpe for the 1 time and for the 2 time probability distributions valid irrespective of the regime of stability of the langevin equations we solve exactly the derived fpes and analyze the aging dynamics by studying analytically the conditional probability distribution we discuss explicitly why the initially conditioned distribution is not sufficient to describe fully out a non markov process as both preparation and observation times have bearing on its dynamics as our analytic procedure can also be applied to linear langevin equations with memory kernels we compare the non markov dynamics of a one delay system with that of a generalized langevin equation with an exponential as well as a power law memory application to a generalization of the green kubo formula is also presented

survival probability of a diffusing particle in the presence of poisson distributed mobile traps the problem of a diffusing particle moving among diffusing traps is analyzed in general space dimension d we consider the case where the traps are initially randomly distributed in space with uniform density ensuremath rho and derive upper and lower bounds for the probability q t averaged over all particle and trap trajectories that the particle survives up to time t we show that for 1l dl 2 the bounds converge asymptotically to give q t ensuremath sim mathrm exp ensuremath ensuremath lambda d t d 2 for 1l dl2 where ensuremath lambda d 2 ensuremath pi d mathrm sin ensuremath pi d 2 4 ensuremath pi d d 2 ensuremath rho and d is the diffusion constant of the traps and that q t ensuremath sim mathrm exp ensuremath 4 ensuremath pi ensuremath rho dt mathrm ln t for d 2 for dg2 bounds can still be derived but they no longer converge for large t for 1l dl 2 these asymptotic forms are independent of the diffusion constant of the particle the results are compared with simulation results obtained using a new algorithm v mehra and p grassberger phys rev e 65 050101 2002 which is described in detail deviations from the predicted asymptotic forms are found to be large even for very small values of q t indicating slowly decaying corrections whose form is consistent with the bounds we also present results in d 1 for the case where the trap densities on either side of the particle are different for this case we can still obtain exact bounds but they no longer converge

finite element methods for the time dependent ginzburg landau model of superconductivity abstract the initial boundary value problem for the time dependent ginzburg landau equations that model the macroscopic behavior of superconductors is considered the convergence of finite dimensional semidiscrete galerkin approximations is studied as is a fully discrete scheme the results of some computational experiments are presented

renormalization group study of thea b diffusion limited reaction thea b diffusion limited reaction with equal initial densitiesa 0 b 0 n0 is studied by means of a field theoretic renormalization group formulation of the problem for dimensiond 2 an effective theory is derived from which the density and correlation functions call be calculated we find the density decays in time as a b sim c sqrt delta dt d 4 ford 4 with n0 c n 0 d 2 wherec is a universal constant andc is nonuniversal the calculation is extended to the case of unequal diffusion constantsd a d b resulting in a new amplitude but the same exponent ford 2 a controlled calculation is not possible but a heuristic argument is presented that the results above give at least the leading term in an e 2 d expansion finally we address reaction zones formed in the steady state by opposing currents ofa andb particles and derive scaling properties

some results on the dynamics of the linear parametric oscillator with general time varying frequency the dynamics of single degree of freedom linear oscillators under parametric excitation of general type periodic or non periodic is considered an analytical approach based on complexification and amplitude phase decomposition of the response is developed leading to an exact set of first order ordinary differential equations governing the amplitude and phase of the motion perturbation schemes for solving this set are developed and the solutions of the derived modulation equation are outlined in addition some results and comments on the stability of the derived solutions are provided

voronoi based finite volume methods optimal voronoi meshes and pdes on the sphere we first develop and analyze a finite volume scheme for the discretization of partial differential equations pdes on the sphere the scheme uses voronoi tessellations of the sphere for a model convection diffusion problem the finite volume scheme is shown to produce first order accurate approximations with respect to a mesh dependent discrete firstderivative norm then we introduce the notion of constrained centroidal voronoi tessellations ccvts of the sphere these are special voronoi tessellation of the sphere for which the generators of the voronoi cells are also the constrained centers of mass with respect to a prescribed density function of the cells after discussing an algorithm for determining ccvt meshes on the sphere we discuss and illustrate several desirable properties possessed by these meshes in particular it is shown that ccvt meshes define very high quality uniform and non uniform meshes on the sphere finally we discuss through some computational experiments the performance of the ccvt meshes used in conjunction with the finite volume scheme for the solution of simple model pdes on the sphere the experiments show for example that the ccvt based finite volume approximations are second order accurate if errors are measured in

spacing distribution functions for the one dimensional point island model with irreversible attachment we study the configurational structure of the point island model for epitaxial growth in one dimension in particular we calculate the island gap and capture zone distributions our model is based on an approximate description of nucleation inside the gaps nucleation is described by the joint probability density p xy n x y which represents the probability density to have nucleation at position x within a gap of size y our proposed functional form for p xy x y describes excellently the statistical behavior of the system we compare our analytical model with extensive numerical simulations our model retains the most relevant physical properties of the system

a finite volume method on general surfaces and its error estimates abstract in this paper we study a finite volume method and its error estimates for the numerical solution of some model second order elliptic partial differential equations defined on a smooth surface the discretization is defined via a surface mesh consisting of piecewise planar triangles and piecewise polygons the optimal error estimates of the approximate solution are proved in both the h 1 and l 2 norms which are of first order and second order respectively under mesh regularity assumptions some numerical tests are also carried out to experimentally verify our theoretical analysis

on the discretization error of parametrized nonlinear equations many applications lead to nonlinear parameter dependent equations h y t y 0 where h y times t to y y 0 in operatorname rge h and the state space y is infinite dimensional while the parameter space t has finite dimension the case dim t 1 is of special interest in connection with continuation methods for this case a general theory is developed which provides for the existence of solution paths of a rather general class of such equations and of their finite dimensional approximations and which allows for an assessment of the error between these paths a principal tool in this analysis is the theory of nonlinear fredholm operators the results cover a more general class of operators than the mildly nonlinear mappings to which other approaches appear to be restricted

probabilistic description of extreme events in intermittently unstable dynamical systems excited by correlated stochastic processes massachusetts institute of technology naval engineering education center grant 3002883706

complex dynamics in initially separated reaction diffusion systems we review recent developments in the study of the diffusion reaction systems of the type a b c in which the reactants are initially separated we consider the case where the a and b particles are initially placed uniformly in euclidean space at x 0 and x 0 respectively we find that whereas for d 2 the mean field exponent characterizes the width of the reaction zone fluctuations are relevant in the one dimensional system we present analytical and numerical results for the reaction rate on fractals and percolation systems at criticality we also study the case where the particles are levy flights in d 1 finally we consider experimentally analytically and numerically the reaction a bstatic c where species a diffuses from a localized source

partial absorption and virtual traps the spatial probability distribution associated with diffusion and attenuation in partially absorbing media is studied an equivalence is established between a system with free diffusion for x 0 and partial absorption for x 0 with a radiation boundary condition at x 0 by exploiting this equivalence it is shown that the effect of a partially absorbing medium in the long time limit is equivalent to that of a perfect virtual trap whose size is smaller than the original absorbing medium for short times however there is substantial penetration of diffusing particles into the absorber the virtual trap approach is readily generalized to higher dimensions this allows one to obtain the density profile of diffusing particles around a partially absorbing spherical trap an unusual crossover between short time penetration and long time trapping occurs in two dimensions the size of the virtual trap is exponentially small in the case of weak absorption corresponding to an absorption time which is exponentially large

centroidal voronoi tessellation based finite element superconvergence in this article a finding on finite element superconvergence is reported the laplacian operator with dirichlet boundary condition is considered the linear finite element solutions have an o h2 0 5 superconvergence in l2 norm at nodes on an almost equilateral triangular mesh generated based on centroidal voronoi tessellation for an arbitrary 2d bounded domain extensive numerical examples are presented to demonstrate the superconvergence property copyright 2008 john wiley sons ltd

numerical approximations of the ginzburg landau models for superconductivity in this paper we review various methods for the numerical approximations of the ginzburg landau models of superconductivity particular attention is given to the different treatment of gauge invariance in both the finite element finite difference and finite volume settings representative theoretical results typical numerical simulations and computational challenges are presented generalizations to other relevant models are also discussed

random walk in a random multiplicative environment we investigate the dynamics of a random walk in a random multiplicative medium this results in a random but correlated multiplicative process for the spatial distribution of random walkers we show how the details of these correlations determine the asymptotic properties of the walk i e the central limit theorem does not apply to these multiplicative processes we also study a periodic source trap medium in which a unit cell contains one source followed byl 1 traps we calculate the asymptotic behavior of the number of particles and determine the conditions for which there is growth or decay in this average number finally we discuss the asymptotic behavior of a random walk in the presence of randomly distributed partially absoprbing traps for this case a temporal regime of purely exponential decay of the density can occur before the asymptotic stretched exponential decay exp at1 3 sets in

a new compact finite difference scheme for solving the complex ginzburg landau equation the complex ginzburg landau equation is often encountered in physics and engineering applications such as nonlinear transmission lines solitons and superconductivity however it remains a challenge to develop simple stable and accurate finite difference schemes for solving the equation because of the nonlinear term most of the existing schemes are obtained based on the crank nicolson method which is fully implicit and must be solved iteratively for each time step in this article we present a fourth order accurate iterative scheme which leads to a tri diagonal linear system in 1d cases we prove that the present scheme is unconditionally stable the scheme is then extended to 2d cases numerical errors and convergence rates of the solutions are tested by several examples

analytical periodic solution and stability assessment of 1 dof parametric systems with time varying stiffness the presented paper deals with an approach to analytical periodic solution and to stability assessment of one degree of freedom linear vibrating systems it is supposed that these systems are excited by the time periodic force and contain time periodic stiffness the periodic green s function determined as a response to a dirac chain of unit impulses repeating with period of excitation is used to transform the equation of motion into the fredholm integral equation with degenerated kernel if the dirac chain is expressed as a fourier series and a limited number of terms is taken into account the solution of the integral equation can also be obtained in a series form it has been found that the real eigenvalues of the system matrix determine the critical values of the fluctuation stiffness parameter the values of this real parameter correspond to the borders of in stability in the plane given by the variation of the angle frequency and of the fluctuation stiffness parameter moreover very interesting property of the system matrix was observed the positive sign of the real valued determinant of the system matrix means the existence of periodic solution system is stable in the opposite case the periodic solution does not exist system is unstable the verification of obtained results was performed on two case studies the floquet method was used to validate the stability assessment presented analytical periodic solution was compared with steady state obtained by the runge kutta continuation a very good agreement was achieved in both cases

anomalous coalescence phenomena under stationary regime condition through thompson s method abstract in this work anomalous coalescence phenomena of type a a a 0 are considered in the case of stationary regime namely the case of an external homogeneous source h of single particles a in the limit of zero external field rate the main purpose here is to pick up new critical exponents in the stationary regime for the phenomenon of anomalous coalescence of bubbles found in one dimension by josserand and rica jr phys rev lett 78 1997 1215 the critical exponents of statical concentration relaxation time and the exponent for concentration decay are evaluated in the limit h 0 zero field rate we show that such critical exponents depend on the parameter physica a 334 2004 335 which characterizes the anomalous diffusion condition in order to explain the anomalous coalescence of jr s work as already made before by the present author physica a 334 2004 335 the parameter must display an explicit dependence on the dimensionality d 4 d for d 2 we use such a result d in order to pick up new critical exponents for the special case of anomalous coalescence found by jr we find 5 4 5 and 1 4 in 1 d we also verify that the known scaling relations among such critical indexes phys rev a 32 2 1985 1129 are obeyed so that in the special case of 2 brownian diffusion condition given to mean field regime we naturally recover those results previously obtained by racz phys rev a 32 2 1985 1129 we go further to give a better explanation for such new results taking into account another exponent physica a 334 2004 335 to explain the behavior of diffusion constant which depends on concentration d d 0 n that is to say we obtain analytically new critical indexes which controll the diffusion behavior d for long time regime in the presence of an external homogeneous source

a probabilistic decomposition synthesis method for the quantification of rare events due to internal instabilities we consider the problem of the probabilistic quantification of dynamical systems that have heavy tailed characteristics these heavy tailed features are associated with rare transient responses due to the occurrence of internal instabilities systems with these properties can be found in a variety of areas including mechanics fluids and waves here we develop a computational method a probabilistic decomposition synthesis technique that takes into account the nature of internal instabilities to inexpensively determine the non gaussian probability density function for any arbitrary quantity of interest our approach relies on the decomposition of the statistics into a non extreme core typically gaussian and a heavy tailed component this decomposition is in full correspondence with a partition of the phase space into a stable region where we have no internal instabilities and a region where non linear instabilities lead to rare transitions with high probability we quantify the statistics in the stable region using a gaussian approximation approach while the non gaussian distribution associated with the intermittently unstable regions of phase space is inexpensively computed through order reduction methods that take into account the strongly nonlinear character of the dynamics the probabilistic information in the two domains is analytically synthesized through a total probability argument the proposed approach allows for the accurate quantification of non gaussian tails at more than 10 standard deviations at a fraction of the cost associated with the direct monte carlo simulations we demonstrate the probabilistic decomposition synthesis method for rare events for two dynamical systems exhibiting extreme events a two degree of freedom system of nonlinearly coupled oscillators and in a nonlinear envelope equation characterizing the propagation of unidirectional water waves

finite volume methods on spheres and spherical centroidal voronoi meshes we study in this paper a finite volume approximation of linear convection diffusion equations defined on a sphere using the spherical voronoi meshes in particular the spherical centroidal voronoi meshes the high quality of spherical centroidal voronoi meshes is illustrated through both theoretical analysis and computational experiments in particular we show that the l2 error of the approximate solution is of quadratic order when the underlying mesh is given by a spherical centroidal voronoi mesh we also demonstrate numerically the high accuracy and the superconvergence of the approximate solutions

statistical behavior of domain systems we study the statistical behavior of two out of equilibrium systems the first one is a quasi one dimensional gas with two species of particles under the action of an external field which drives each species in opposite directions the second one is a one dimensional spin system with nearest neighbor interactions also under the influence of an external driving force both systems show a dynamical scaling with domain formation the statistical behavior of these domains is compared with models based on the coalescing random walk and the interacting random walk we find that the scaling domain size distribution of the gas and the spin systems is well fitted by the wigner surmise which lead us to explore a possible connection between these systems and the circular orthogonal ensemble of random matrices however the study of the correlation function of the domain edges shows that the statistical behavior of the domains in both gas and spin systems is not completely well described by a circular orthogonal ensemble nor it is by other models proposed such as the coalescing random walk and the interacting random walk nevertheless we find that a simple model of independent intervals describes more closely the statistical behavior of the domains formed in these systems

finite dimensional approximation of nonlinear problems in the first two papers of this series 4 5 we have studied a general method of approximation of nonsingular solutions and simple limit points of nonlinear equations in a banach space we derive here general approximation results of the branches of solutions in the neighborhood of a simple bifurcation point the abstract theory is applied to the galerkin approximation of nonlinear variational problems and to a mixed finite element approximation of the von karman equations

fluctuation effects and multiscaling of the reaction diffusion front for a b to oe we consider the properties of the diffusion controlled reaction a b to oe in the steady state where fixed currents of a and b particles are maintained at opposite edges of the system using renormalization group methods we explicitly calculate the asymptotic forms of the reaction front and particle densities as expansions in jd 1 mod x mod d 1 1 where j are the equal applied currents and d the equal diffusion constants for the asymptotic densities of the minority species we find in addition to the expected exponential decay fluctuation induced power law tails which for d 2 we find similar power laws with mu d 3 but with non universal coefficients logarithmic corrections occur in d 2 these results imply that in the time dependent case with segregated initial conditions the moments integral mod x mod qr x t dx fail to satisfy simple scaling for q mu 1 finally it is shown that the fluctuation induced wandering of the position of the reaction front centre may be neglected for large enough systems

finite element approximation of a periodic ginzburg landau model for type ii superconductors we consider efficient finite element algorithms for the computational simulation of type ii superconductors the algorithms are based on discretizations of a periodic ginzburg landau model periodicity is defined with respect to a non orthogonal lattice that is not necessarily aligned with the coordinate axes also the primary dependent variables employed in the model satisfy non standard quasi periodic boundary conditions after introducing the model we define finite element schemes derive error estimates of optimal order and present the results of some numerical calculations for a similar quality of simulation the resulting algorithms seem to be significantly less costly than are previously used numerical approximation methods

a non homogeneous hill s equation the existence of periodic solutions for a forced hill s equation is proved the proof is then extended to the case of a non homogeneous matrix valued hill s equation under the stated conditions using lyapunov s criteria proc ams 13 1962 601 hill s equation interscience publishers new york 1966 some results on the stability oh hill s equation are obtained

discrete gauge invariant approximations of a time dependent ginzburg landau model of superconductivity we present here a mathematical analysis of a nonstandard difference method for the numerical solution of the time dependent ginzburg landau models of superconductivity this type of method has been widely used in numerical simulations of the behavior of superconducting materials we also illustrate some of their nice properties such as the gauge invariance being retained in discrete approximations and the discrete order parameter having physically consistent pointwise bound

spectral signature of the pitchfork bifurcation liouville equation approach the time evolution of probability densities of one dimensional nonlinear vector fields is studied using a liouville equation approach it is shown that the liouville operator admits a discrete spectrum of eigenvalues of decaying type if the vector field is far from bifurcation the associated right and left eigenvectors are explicitly constructed for simple models and shown to be distributions rather than regular functions on the other hand the spectrum of the liouville operator may become continuous at the bifurcation point a phenomenon illustrated explicitly in the paper in the case of the pitchfork bifurcation the relationship between the spectral decompositions of the liouville and of the fokker planck equations is discussed in particular the spectral decompositions constructed here for the liouville equation are obtained as the noiseless limit of the well known spectral decompositions of the fokker planck equation of the associated stochastic process

on periodic orbits of nonlinear dynamical systems with many degrees of freedom abstract the purpose of this article is to extend our investigation to periodic orbits of nonlinear dynamical systems with many degrees of freedom the method of generalized averaging and the indicatrix method are developed an example is studied to illustrate our techniques the analytical expressions are in good agreement with numerical results computed by the indicatrix method for this example

subharmonic solutions of hill s equation in this article we use the method of reflecting function to discuss the qualitative behavior of the solutions of hill s equations we give the necessary and sufficient conditions for hill s equation to have some reflecting functions and apply the obtained results to discuss the existence of periodic and subharmonic solutions of non homogeneous hill s equation

reaction diffusion theory in the presence of an attractive harmonic potential problems involving the capture of a moving entity by a trap occur in a variety of physical situations the moving entity being an electron an excitation an atom a molecule a biological object such as a receptor cluster a cell or even an animal such as a mouse carrying an epidemic theoretical considerations have almost always assumed that the particle motion is translationally invariant we study here the case when that assumption is relaxed in that the particle is additionally subjected to a harmonic potential this tethering to a center modifies the reaction diffusion phenomenon using a smoluchowski equation to describe the system we carry out a study which is explicit in one dimension but can be easily extended for arbitrary dimensions interesting features emerge depending on the relative location of the trap the attractive center and the initial placement of the diffusing particle

nearest neighbor distances in diffusion controlled reactions modelled by a single mobile trap we consider a system consisting of an infinite number of identical particles on a latttce initially uniformly distributed which diffuse in the presence of a single mobile trap and ask for the time dependent behavior of the distance of the trap from the nearest particle this quantity is a measure of the tendency of the system to self segregate we show by a simulation incorporating the exact enumeration method that in one dtmension the expected distance l t scales as l t t as t 0 where the exponent a depends only on the ratio of the diffusion constants a heuristic expression for a ts suggested analogous to a rigorous exponent found by ben avraham for a similar but not identical problem the flux into the trap is found to vary as t t independent of the dtffusion constants

one dimensional δ function potential and the radiation boundary condition in a recent paper 1 datta and jayannavar d j suggest a g function potential model to describe a single imperfect trap in one dimension as an alternative to the traditional radiation boundary condition rbc approach 2 using their model they re study some of the statistical properties of nearest neighbor distance at the trap this distance is one of the possible measures of self segregation in trapping reactions a topic of current great interest 3 10 in the rbc description actual particles are restricted to one side of the trap of which some are absorbed at the trap while others are reflected back as dj point out such a constraint is removed in their treatment diffusing particles impinging on the trap can partially be absorbed and partially move away to either side of the trap dj obtain different results for the nearest neighbor distance statistics when particles are initially distributed along the positive semi infinite space but for initial uniform distribution throughout the infinite line their results are equivalent to those obtained using the rbc 5 with redefined parameters although dj do explain this similarity by physical symmetry arguments they claim that the equivalence is a coincidence in this paper we show how this equivalence can be obtained explicitly we also address the question of the appropriateness of the two approaches to describe partial absorbing traps in various physical systems

on the use of local maximum entropy approximants for cahn hilliard phase field models in 2d domains and on surfaces abstract we apply a local maximum entropy lme approximation scheme to the fourth order phase field model for the traditional cahn hilliard theory the discretization of the cahn hilliard equation by galerkin method requires at least c 1 continuous basis functions this requirement can be fulfilled by using lme shape functions which are c continuous in this case the primal variational formulations of the fourth order partial differential equation are well defined and integrable hence there is no need to split the fourth order partial differential equation into two second order partial differential equations this splitting scheme is a common practice in mixed finite element formulations with c 0 continuous lagrange shape functions furthermore we use a general and simple numerical method such as statistical manifold learning technique that allows dealing with general point set surfaces avoiding a global parameterization which can be applied to tackle surfaces of complex geometry and topology and to solve cahn hilliard equation on general surfaces finally the flexibility and robustness of the presented methodology is demonstrated for several representative numerical examples

statistical properties of the distance between a trapping center and a uniform density of diffusing particles in two dimensions several analyses of self segregation properties of reaction diffusion systems in low dimensions have been based on a simplified model in which an initially uniform concentration of point particles is depleted by reaction with an immobilized trap a measure of selfsegregation in this system is the distance of the trap from the nearest untrapped particle in one dimension the average of this distance has been shown to increase at a rate proportional to t 4 we show that this rate in a two dimensional system is asymptotically proportional to in t 2 and that the concentration profile in the neighborhood of the trap is proportional to in r ln t

statistical properties of the nearest neighbour distance at a single imperfect trap we consider a one dimensional system consisting of a single partially absorbing trap in the presence of diffusing particles the probability density function of the closest distance between the trap and the untrapped particle is calculated exactly in particular we show the equivalence between our result and that of earlier treatments with the radiation boundary condition only when the initial distribution of particles is uniform throughout the infinite space our result differs from the earlier treatments when the initial particle distribution is non uniform

capture zone areas in submonolayer nucleation effects of dimensionality and short range interactions simulation results for the asymptotic scaled capture zone distribution czd for the case of irreversible nucleation and growth of point islands are presented for substrate dimension d 1 2 3 and 4 and compared with a recent conjecture based on the wigner distribution poor agreement is found between the predicted wigner distributions and the asymptotic czd in the limit of infinite df corresponding to the ratio of monomer hopping rate d to deposition rate f our results also indicate that for d 2 and 3 the asymptotic czd for point islands is independent of model details and dimension however for d 1 and d 4 the resulting distribution is significantly more sharply peaked we also find that in contrast to the island size distribution for which mean field like behavior is observed in d 3 and above the asymptotic czd is significantly broadened by fluctuations even in d 4

first exit times of harmonically trapped particles a didactic review we revise the classical problem of characterizing first exit times of a harmonically trapped particle whose motion is described by a one or multidimensional ornstein uhlenbeck process we start by recalling the main derivation steps of a propagator using langevin and fokker planck equations the mean exit time the moment generating function and the survival probability are then expressed through confluent hypergeometric functions and thoroughly analyzed we also present a rapidly converging series representation of confluent hypergeometric functions that is particularly well suited for numerical computation of eigenvalues and eigenfunctions of the governing fokker planck operator we discuss several applications of first exit times such as the detection of time intervals during which motor proteins exert a constant force onto a tracer in optical tweezers single particle tracking experiments adhesion bond dissociation under mechanical stress characterization of active periods of trend following and mean reverting strategies in algorithmic trading on stock markets relation to the distribution of first crossing times of a moving boundary by brownian motion some extensions are described including diffusion under quadratic double well potential and anomalous diffusion

trace formula for noisy flows a trace formula is derived for the fokker planck equation associated with ito stochastic differential equations describing noisy time continuous nonlinear dynamical systems in the weak noise limit the trace formula provides estimations of the eigenvalues of the fokker planck operator on the basis of the pollicott ruelle resonances of the noiseless deterministic system which is assumed to be non bifurcating at first order in the noise amplitude the effect of noise on a periodic orbit is given in terms of the period and the derivative of the period with respect to the pseudo energy of the onsager machlup freidlin wentzell scheme

analysis and convergence of a covolume approximation of the ginzburg landau model of superconductivity in this paper we present the mathematical analysis of a covolume method for the approximations of the ginzburg landau gl model for superconductivity a nice feature of this approach is that the gauge invariance properties are retained in discrete approximations based on triangular grids we also use properties of discrete vector fields to study issues such as the gauge choices and their enforcement

localized partial traps in diffusion processes and random walks reaction diffusion equations in which the reaction is described by a sink term consisting of a sum of delta functions are studied it is shown that the laplace transform of the reactive green s function can be analytically expressed in terms of the green s function for diffusion in the absence of reaction moreover a simple relation between the green s functions satisfying the radiation boundary condition and the reflecting boundary condition is obtained several applications are presented and the formalism is used to establish the relationship between the time dependent geminate recombination yield and the bimolecular reaction rate for diffusion influenced reactions finally an analogous development for lattice random walks is presented