a conservative free energy dissipating and positivity preserving finite difference scheme for multi dimensional nonlocal fokker planck equation abstract in this work we design and analyze a conservative positivity preserving and free energy dissipating finite difference method for the multi dimensional nonlocal fokker planck fp equation based on a non logarithmic landau transformation a central differencing spatial discretization using harmonic mean approximations is developed both forward and backward euler discretizations in time are employed to derive an explicit scheme and a linearized semi implicit scheme respectively three desired properties that are possessed by analytical solutions i mass conservation ii free energy dissipation and iii positivity are proved to be maintained at discrete level remarkably numerical analysis demonstrates that the semi implicit time discretization ensures the property of positivity preserving unconditionally due to the advantages brought by the harmonic mean approximations an estimate on the upper bound of the condition number of the resulting coefficient matrix is further established for the semi implicit scheme extensive numerical tests are performed to validate aforementioned properties numerically

lp theory for the multidimensional aggregation equation we consider well posedness of the aggregation equation tu div uv 0 v k u with initial data in input amssym cal p 2 rm bbb r d rm cap l p bbb r d in dimensions 2 and higher we consider radially symmetric kernels where the singularity at the origin is of order x 2 d and prove local well posedness in input amssym cal p 2 bbb r d rm cap l p bbb r d for sufficiently large p ps in the special case of k x x the exponent ps d d 1 is sharp for local well posedness in that solutions can instantaneously concentrate mass for initial data in input amssym cal p 2 bbb r d rm cap l p bbb r d with p ps we also give an osgood condition on the potential k x that guarantees global existence and uniqueness in input amssym cal p 2 bbb r d rm cap l p bbb r d 2010 wiley periodicals inc

a cartesian grid embedded boundary method for hyperbolic conservation laws we present a second order godunov algorithm to solve time dependent hyperbolic systems of conservation laws on irregular domains our approach is based on a formally consistent discretization of the conservation laws on a finite volume grid obtained from intersecting the domain with a cartesian grid we address the small cell stability problem associated with such methods by hybridizing our conservative discretization with a stable nonconservative discretization at irregular control volumes and redistributing the difference in the mass increments to nearby cells in a way that preserves stability and local conservation the resulting method is second order accurate in l 1 for smooth problems and is robust in the presence of large amplitude discontinuities intersecting the irregular boundary

a sharp interface immersed boundary method with improved mass conservation and reduced spurious pressure oscillations a method for reducing the spurious pressure oscillations observed when simulating moving boundary flow problems with sharp interface immersed boundary methods ibms is proposed by first identifying the primary cause of these oscillations to be the violation of the geometric conservation law near the immersed boundary we adopt a cut cell based approach to strictly enforce geometric conservation in order to limit the complexity associated with the cut cell method the cut cell based discretization is limited only to the pressure poisson and velocity correction equations in the fractional step method and the small cell problem tackled by introducing a virtual cell merging technique the method is shown to retain all the desirable properties of the original finite difference based ibm while at the same time reducing pressure oscillations for moving boundaries by roughly an order of magnitude

non gaussian garch option pricing models and their diffusion limits this paper investigates the weak convergence of general non gaussian garch models together with an application to the pricing of european style options determined using an extended girsanov principle and a conditional esscher transform as the pricing kernel candidates applying these changes of measure to asymmetric garch models sampled at increasing frequencies we obtain two risk neutral families of processes which converge to different bivariate diffusions which are no longer standard hull white stochastic volatility models regardless of the innovations used the garch implied diffusion limit based on the esscher transform can be obtained by applying the minimal martingale measure under the physical measure however we further show that for skewed garch driving noise the risk neutral diffusion limit of the extended girsanov principle exhibits a non zero market price of volatility risk which is proportional to the market price of the equity risk where the constant of proportionality depends on the skewness and kurtosis of the underlying distribution our theoretical results are further supported by numerical simulations and a calibration exercise to observed market quotes

obstacle and predator avoidance in a model for flocking abstract the modeling and investigation of the dynamics and configurations of animal groups is a subject of growing attention in this paper we present a deterministic continuum model of flocking and use it to investigate the linearized response of an infinite extent flock to an obstacle or an attacking predator we show that the steady state flock response is in the form of density disturbances that resemble mach cones whose configuration is determined by the anisotropic propagation of waves through the flock we analytically and numerically test relations that predict the mach wedge angles disturbance heights and wake widths we find that these expressions are insensitive to many of the parameters of the model

on the problem of reconstructing an unknown topology via locality properties of the wiener filter determining interrelatedness structure of various entities from multiple time series data is of significant interest to many areas knowledge of such a structure can aid in identifying cause and effect relationships clustering of similar entities identification of representative elements and model reduction the majority of existing results are based on correlation based indices which effectively assume a static relationship between the time series data and are not suitable for detecting interrelatedness when the time series are dynamically related or when the time series involve loops in this paper a methodology for identifying the interrelatedness structure of dynamically related time series data is presented that also allows for the presence of loops in the connectivity structure a linear dynamic graph model is presented where it is assumed that each time series data is the sum of an independent stochastic noise source and a dynamically weighted sum of other time series data a link is assumed to be present between two time series if the weight of a time series which is a linear time invariant filter is nonzero in the formation of the other reconstruction of the link connectivity structure under various scenarios is considered it is shown that when the linear dynamic graph is allowed to admit non causal weights then the links structure can be recovered with the possibility of identifying spurious connections however it is shown that the spurious links remain local where a spurious link is restricted to be within one hop of a true link furthermore strategies for exact reconstruction of the link structure when the weights are restricted to be causal are developed the main tools for determining the network topology are based on variations of wiener filtering a significant insight provided by the article is that in the class of network models considered in the paper the wiener filter estimating a stochastic process based on other processes remains local in the sense that the wiener filter utilizes only measurements local to the node being estimated

a new interaction potential for swarming models abstract we consider a self propelled particle system which has been used to describe certain types of collective motion of animals such as fish schools and bird flocks interactions between particles are specified by means of a pairwise potential repulsive at short ranges and attractive at longer ranges the exponentially decaying morse potential is a typical choice and is known to reproduce certain types of collective motion observed in nature particularly aligned flocks and rotating mills we introduce a class of interaction potentials that we call quasi morse for which flock and rotating mills states are also observed numerically however in that case the corresponding macroscopic equations allow for explicit solutions in terms of special functions with coefficients that can be obtained numerically without solving the particle evolution we compare the obtained solutions with long time dynamics of the particle systems and find a close agreement for several types of flock and mill solutions

cut element based immersed boundary method for moving geometries in compressible liquid flows with cavitation the conservative immersed interface method for representing complex immersed solid boundaries or phase interfaces on cartesian grids is improved and extended to allow for the simulation of weakly compressible fluid flows through moving geometries we demonstrate that an approximation of moving interfaces by a level set field results in unphysical oscillations in the vicinity of sharp corners when dealing with weakly compressible fluids such as water by introducing an exact reconstruction of the cut cell properties directly based on a surface triangulation of the immersed boundary we are able to recover the correct flow evolution free of numerical artifacts the new method is based on cut elements it provides sub cell resolution of the geometry and handles flows through narrow closing or opening gaps in a straightforward manner we validate our method with canonical flows around oscillating cylinders we demonstrate that the method allows for an accurate prediction of flows around moving obstacles in weakly compressible liquid flows with cavitation effects in particular we show that the cavitating flow through a closing fuel injector control valve which is an example for a complex application with interaction of stationary and moving parts can be predicted by the method

finite size scaling analysis and dynamic study of the critical behavior of a model for the collective displacement of self driven individuals the vicsek model vm t vicsek et al phys rev lett 75 1226 1995 for the description of the collective behavior of self driven individuals assumes that each of them adopts the average direction of movement of its neighbors perturbed by an external noise a second order transition between a state of ordered collective displacement low noise limit and a disordered regime high noise limit was found early on however this scenario has recently been challenged by gr egory and chat e g gr egory and h chat e phys rev lett 92 025702 2004 who claim that the transition of the vm may be of first order by performing extensive simulations of the vm which are analyzed by means of both finite size scaling methods and a dynamic scaling approach we unambiguously demonstrate the critical nature of the transition furthermore the complete set of critical exponents of the vm in d 2 dimensions is determined by means of independent methods i e stationary and dynamic measurements we provide two tests showing that the standard hyperscaling relationship d ensuremath nu ensuremath 2 ensuremath beta ensuremath gamma holds where ensuremath beta ensuremath nu and ensuremath gamma are the order parameter correlation length and susceptibility critical exponents respectively furthermore we established that at criticality the correlation length grows according to ensuremath xi ensuremath sim t 1 z with z ensuremath simeq 1 27 3 independently of the degree of order of the initial configuration in marked contrast with the behavior of the xy model

augmented garch p q process and its diffusion limit abstract a family of parametric garch models defined in terms of an auxiliary process and referred to as the augmented garch process is proposed the strict stationarity of the augmented garch process is characterized and this process is shown to contain many existing parametric garch models the augmented garch process can serve as a general alternative for lagrange multiplier test of many existing garch specifications the diffusion limit of the augmented garch process is shown to contain many bivariate diffusion processes that are commonly used for modeling stochastic volatility in the finance literature this convergence result generalizes that of nelson 1990a to cover a substantially larger class of garch 1 1 models and also extends to the garch p q specification the augmented garch process can be used as a direct approximation to the stochastic volatility models or as the score generator in the efficient method of moments gallant and tauchen 1996 estimation of these models

well posedness theory for aggregation sheets in this paper we consider distribution solutions to the aggregation equation rho t mathrm div rho mathbf u 0 mathbf u nabla v rho in mathbb r d where the density concentrates on a co dimension one manifold we show that an evolution equation for the manifold itself completely determines the dynamics of such solutions we refer to such solutions aggregation sheets when the equation for the sheet is linearly well posed we show that the fully non linear evolution is also well posed locally in time for the class of bi lipschitz surfaces moreover we show that if the initial sheet is c1 then the solution itself remains c1 as long as it remains lipschitz lastly we provide conditions on the kernel g s frac mathrm d v mathrm d s that guarantee the solution remains a bi lipschitz surface globally in time and construct explicit solutions that either collapse or blow up in finite time when these conditions fail

accurate immersed boundary reconstructions for viscous flow simulations cells crossed by the immersed boundary is treated a novel interpolation method based on momentum balance and massconservationbasedon finitevolumemethodareintroducedtocorrectlypredictthevelocityandpressure fieldin the vicinity of the boundaries the assessment of the solution quality is based on calculations of taylor decaying vortices examples of laminar and turbulent flow calculations are also reported

some weak flocking models and its application to target tracking abstract in this paper we introduce a new concept named the weak flocking behavior that is if the complex system has the weak flocking property the agents do not need have the same velocities to keep together when t this shows the biggest difference between weak flocking behavior and the flocking behavior the advantage of this concept is that its practical significance the agents don t need always keep in step in lots of the complex systems but they still keep together then we propose a weak flocking model with two agents and study two different mechanisms for this model moreover we also consider a system with n agents under the weak link mechanism the system with n agents can keep weak flocking

existence properties and trajectory specification of generalised multi agent flocking a class of flocking algorithms for second order multi agent networks with single virtual leaders are developed by including more weighting and navigating matrices in this paper flocking existence

promoting collective motion of self propelled agents by discarding short range interactions abstract we study the collective motion of self propelled agents with the restricted view the field of view of each agent is an annulus bounded by the outer radius r and inner radius r where is a tunable parameter we find that there exists an optimal value of leading to the highest degree of direction consensus this phenomenon indicates that there exists superfluous communication in the collective motion of self propelled agents and short range interactions hinder the direction consensus of the system the value of optimal decreases as the absolute velocity increases while it increases as the outer radius r and the system size increase for a fixed value of direction consensus is enhanced when the absolute velocity is small the outer radius or the system size is large

streaming disruptive interference and power law behavior in the exit dynamics of confined pedestrians we analyze the exit dynamics of pedestrians who are initially confined in a room pedestrians are modeled as cellular automata and compete to escape via a known exit at the soonest possible time a pedestrian could move forward backward left or right within each iteration time depending on adjacent cell vacancy and in accordance with simple rules that determine the compulsion to move and physical capability relative to his neighbors the arching signatures of jamming were observed and the pedestrians exited in bursts of various sizes power law behavior is found in the burst size frequency distribution for exit widths w greater than one cell dimension w 1 the slope of the power law curve varies with w from 1 3092 w 2 to 1 0720 w 20 streaming which is a diffusive behavior arises in large burst sizes and is more likely in a single exit room with w 1 and leads to a counterintuitive result wherein an average exit throughput q is obtained that is higher than with w 2 3 or 4 for a two exit room w 1 q is not greater than twice the yield of a single exit room if the doors are not separated far enough 4w q becomes even significantly less due to a collective slow down that emerges among pedestrians crossing in each other s path disruptive interference effect for the same w and door number q is also higher with relaxed pedestrians than with anxious ones

arch models as diffusion approximations abstract this paper investigates the convergence of stochastic difference equations e g arch to stochastic differential equations as the length of the discrete time intervals between observations goes to zero these results are applied to the garch 1 1 model of bollerslev 1986 and to the ar 1 exponential arch model of nelson 1989 in their continuous time limits the conditional variance processes in these models have stationary distributions that are inverted gamma and lognormal respectively in addition a class of diffusion approximations based on the exponential arch model is developed

global regularity for vortex patches we present a proof of chemin s 4 result which states that the boundary of a vortex patch remains smooth for all time if it is initially smooth

a kinetic flocking model with diffusion we study the stability of the equilibrium states and the rate of convergence of solutions towards them for the continuous kinetic version of the cucker smale flocking in presence of diffusion whose strength depends on the density this kinetic equation describes the collective behavior of an ensemble of organisms animals or devices which are forced to adapt their velocities according to a certain rule implying a final configuration in which the ensemble flies at the mean velocity of the initial configuration our analysis takes advantage both from the fact that the global equilibrium is a maxwellian distribution function and on the contrary to what happens in the cucker smale model ieee trans autom control 52 852 862 2007 the interaction potential is an integrable function precise conditions which guarantee polynomial rates of convergence towards the global equilibrium are found

regular article an accurate cartesian grid method for viscous incompressible flows with complex immersed boundaries a cartesian grid method has been developed for simulating two dimensional unsteady viscous incompressible flows with complex immersed boundaries a finite volume method based on a second order accurate central difference scheme is used in conjunction with a two step fractional step procedure the key aspects that need to be considered in developing such a solver are imposition of boundary conditions on the immersed boundaries and accurate discretization of the governing equation in cells that are cut by these boundaries a new interpolation procedure is presented which allows systematic development of a spatial discretization scheme that preserves the second order spatial accuracy of the underlying solver the presence of immersed boundaries alters the conditioning of the linear operators and this can slow down the iterative solution of these equations the convergence is accelerated by using a preconditioned conjugate gradient method where the preconditioner takes advantage of the structured nature of the underlying mesh the accuracy and fidelity of the solver is validated by simulating a number of canonical flows and the ability of the solver to simulate flows with very complicated immersed boundaries is demonstrated

a fast algorithm for convolution integrals with space and time variant kernels mathematical models in many fields of the physical sciences involve nonlocal terms which are formally similar to convolution integrals we show that it is possible to approximate a particular class of such integrals which by themselves are not convolutions as a linear combination of convolution integrals allowing for their efficient numerical computation as an o n log n process

gas kinetic derivation of navier stokes like traffic equations macroscopic traffic models have recently been severely criticized to base on lax analogies only and to have a number of deficiencies therefore this paper shows how to construct a logically consistent fluid dynamic traffic model from basic laws for the acceleration and interaction of vehicles these considerations lead to the gas kinetic traffic equation of paveri fontana its stationary and spatially homogeneous solution implies equilibrium relations for the fundamental diagram the variance density relation and other quantities which are partly difficult to determine empirically paveri fontana s traffic equation allows the derivation of macroscopic moment equations which build a system of non closed equations this system can be closed by the well proved method of chapman and enskog which leads to euler like traffic equations in zeroth order approximation and to navier stokes like traffic equations in first order approximation the latter are finally corrected for the finite space requirements of vehicles it is shown that the resulting model is able to withstand the above mentioned criticism

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

state transitions and the continuum limit for a 2d interacting self propelled particle system we study a class of swarming problems wherein particles evolve dynamically via pairwise interaction potentials and a velocity selection mechanism we find that the swarming system undergoes various changes of state as a function of the self propulsion and interaction potential parameters in this paper we utilize a procedure which connects a class of individual based models to their continuum formulations and determine criteria for the validity of the latter h stability of the interaction potential plays a fundamental role in determining both the validity of the continuum approximation and the nature of the aggregation state transitions we perform a linear stability analysis of the continuum model and compare the results to the simulations of the individual based one c 2007 elsevier b v all rights reserved

numerical simulation of nonlinear continuity equations by evolving diffeomorphisms abstract in this paper we present a numerical scheme for nonlinear continuity equations which is based on the gradient flow formulation of an energy functional with respect to the quadratic transportation distance it can be applied to a large class of nonlinear continuity equations whose dynamics are driven by internal energies given external potentials and or interaction energies the solver is based on its variational formulation as a gradient flow with respect to the wasserstein distance positivity of solutions as well as energy decrease of the semi discrete scheme are guaranteed by its construction we illustrate this property with various examples in spatial dimension one and two

self organization in systems of self propelled particles we investigate a discrete model consisting of self propelled particles that obey simple interaction rules we show that this model can self organize and exhibit coherent localized solutions in one and in two dimensions in one dimension the self organized solution is a localized flock of finite extent in which the density abruptly drops to zero at the edges in two dimensions we focus on the vortex solution in which the particles rotate around a common center and show that this solution can be obtained from random initial conditions even in the absence of a confining boundary furthermore we develop a continuum version of our discrete model and demonstrate that the agreement between the discrete and the continuum model is excellent

multiparticle random walks on a deformable medium multiparticle random walks on a deformable medium have been investigated in 2 1 dimensions the time evolution of the particle distribution is studied the results show that the randomly distributed particles in the beginning will be self organized into a cluster pattern in the intermediate stage and then return to the random distribution pattern in the late stage the dependence of the clustering degree on the stiffness parameter of medium ensuremath alpha stability parameter of systems ensuremath beta and average particle density ensuremath rho 0 is also investigated there exists an optimal clustering stability ensuremath beta p at which the system has the strongest clustering ability and corresponds to a maximum clustering coefficient ensuremath gamma p the dependence of the optimal clustering coefficient ensuremath gamma p on the stiffness ensuremath alpha and particle density ensuremath rho 0 is obtained and the landscape of the medium generated by particles is also investigated

a cut cell finite volume finite element coupling approach for fluid structure interaction in compressible flow we present a loosely coupled approach for the solution of fluid structure interaction problems between a compressible flow and a deformable structure the method is based on staggered dirichlet neumann partitioning the interface motion in the eulerian frame is accounted for by a conservative cut cell immersed boundary method the present approach enables sub cell resolution by considering individual cut elements within a single fluid cell which guarantees an accurate representation of the time varying solid interface the cut cell procedure inevitably leads to non matching interfaces demanding for a special treatment a mortar method is chosen in order to obtain a conservative and consistent load transfer we validate our method by investigating two dimensional test cases comprising a shock loaded rigid cylinder and a deformable panel moreover the aeroelastic instability of a thin plate structure is studied with a focus on the prediction of flutter onset finally we propose a three dimensional fluid structure interaction test case of a flexible inflated thin shell interacting with a shock wave involving large and complex structural deformations

phase transitions in self driven many particle systems and related non equilibrium models a network approach we investigate the conditions that produce a phase transition from an ordered to a disordered state in a family of models of two dimensional elements with a ferromagnetic like interaction this family is defined to contain under the same framework among others the xy model and the self driven particles model introduced by vicsek et al each model is distinguished only by the rules that determine the set of elements with which each element interacts we propose a new member of the family the vectorial network model in which a given fraction of the elements interact through direct random connections this model is analogous to an xy system on a network and as such can be of interest for a wide range of problems it captures the main aspects of the interaction dynamics that produce the phase transition in other models of the family the network approach allows us to show analytically the existence of a phase transition in this vectorial network model and to compute its relevant parameters for the case in which all elements are randomly connected finally we study numerically the conditions required for a phase transition to exist for different members of the family our results show that a qualitatively equivalent phase transition appears whenever even a small amount of long range interactions are present or built over time regardless of other equilibrium or non equilibrium properties of the system

boundary condition enforced immersed boundary lattice boltzmann flux solver for thermal flows with neumann boundary conditions a boundary condition enforced immersed boundary lattice boltzmann flux solver is proposed in this work for effective simulation of thermal flows with neumann boundary conditions in this method two auxiliary layers of lagrangian points are introduced and respectively placed inside and outside of the solid body on which the temperature corrections related to the heat source are set as unknowns to effectively consider the fluid boundary interaction these unknowns are expressed as algebraic summations of the temperature correction on eulerian points which are in turn obtained from biased distributions of unknown temperature corrections on the immersed boundary by enforcing the temperature gradient at the solid boundary being equal to that approximated by the corrected temperature field a set of algebraic equations are formed and solved to obtain all the unknowns simultaneously they are then distributed biasedly to the inner region of the auxiliary layer so that the diffusion from the smooth delta function can be reduced substantially in addition the solutions of the flow and temperature fields are obtained by the thermal lattice boltzmann flux solver with the second order of accuracy the proposed method is well validated through its applications to simulate several benchmarks of natural forced and mixed convection problems it has been demonstrated that the present solver has about 1 724 order of accuracy and the error between the present result and theoretical value for the temperature gradient on the solid surface is in the order of 10 13 which indicates that the proposed method is able to satisfy the neumann boundary condition accurately

flocks herds and schools a quantitative theory of flocking we present a quantitative continuum theory of flocking the collective coherent motion of large numbers of self propelled organisms in agreement with everyday experience our model predicts the existence of an ordered phase of flocks in which all members of even an arbitrarily large flock move together with the same mean velocity stackrel ensuremath rightarrow v ensuremath ne 0 this coherent motion of the flock is an example of spontaneously broken symmetry no preferred direction for the motion is picked out a priori in the model rather each flock is allowed to and does spontaneously pick out some completely arbitrary direction to move in by analyzing our model we can make detailed quantitative predictions for the long distance long time behavior of this broken symmetry state the goldstone modes associated with this spontaneously broken rotational symmetry are fluctuations in the direction of motion of a large part of the flock away from the mean direction of motion of the flock as a whole these goldstone modes mix with modes associated with conservation of bird number to produce propagating sound modes these sound modes lead to enormous fluctuations of the density of the flock far larger at long wavelengths than those in e g an equilibrium gas our model is similar in many ways to the navier stokes equations for a simple compressible fluid in other ways it resembles a relaxational time dependent ginsburg landau theory for an n d component isotropic ferromagnet in spatial dimensions dg4 the long distance behavior is correctly described by a linearized theory and is equivalent to that of an unusual but nonetheless equilibrium model for spin systems for dl4 nonlinear fluctuation effects radically alter the long distance behavior making it different from that of any known equilibrium model in particular we find that in d 2 where we can calculate the scaling exponents exactly flocks exhibit a true long range ordered spontaneously broken symmetry state in contrast to equilibrium systems which cannot spontaneously break a continuous symmetry in d 2 the mermin wagner theorem we make detailed predictions for various correlation functions that could be measured either in simulations or by quantitative imaging of real flocks we also consider an anisotropic model in which the birds move preferentially in an easy e g horizontal plane and make analogous but quantitatively different predictions for that model as well for this anisotropic model we obtain exact scaling exponents for all spatial dimensions including the physically relevant case d 3

refined blowup criteria and nonsymmetric blowup of an aggregation equation abstract we consider an aggregation equation in r d d 2 with fractional dissipation u t u k u u where 0 0 1 and k x e x we prove a refined blowup criteria by which the global existence of solutions is controlled by its l x q norm for any d d 1 q we prove the finite time blowup of solutions for a general class of nonsymmetric initial data the argument presented works for both the inviscid case 0 and the supercritical case 0 and 0 1 additionally we present new proofs of blowup which does not use free energy arguments

stochastic flocking dynamics of the cucker smale model with multiplicative white noises we present a strong asymptotic stochastic flocking estimate for the stochastically perturbed cucker smale model we characterize a form of multiplicative white noises and present sufficient conditions on the control parameters to guarantee the almost sure exponential convergence toward constant equilibrium states when the strength of multiplicative noises is sufficiently large we show that the strong stochastic flocking occurs even for negative communication weights

nonlocal aggregation models a primer of swarm equilibria biological aggregations such as fish schools bird flocks bacterial colonies and insect swarms have characteristic morphologies governed by the group members intrinsic social interactions with each other and by their interactions with the external environment starting from a simple discrete model treating individual organisms as point particles we derive a nonlocal partial differential equation describing the evolving population density of a continuum aggregation to study equilibria and their stability we use tools from the calculus of variations in one spatial dimension and for several choices of social forces external forces and domains we find exact analytical expressions for the equilibria these solutions agree closely with numerical simulations of the underlying discrete model the analytical solutions provide a sampling of the wide variety of equilibrium configurations possible within our general swarm modeling framework and include features such as spatial localization with compact su

hierarchical cluster structures in a one dimensional swarm oscillator model swarm oscillator model derived by one of the authors tanaka where interacting motile elements form various kinds of patterns is investigated we particularly focus on the cluster patterns in one dimensional space we mathematically derive all static and stable configurations in final states for a particular but a large set of parameters in the derivation we introduce renormalized expression of this model we find that the static final states are hierarchical cluster structures in which a cluster consists of smaller clusters in a nesting manner

a two dimensional conservation laws scheme for compressible flows with moving boundaries the generalized riemann problem grp scheme for the hydrodynamic conservation laws is extended to a two dimensional moving boundary tracking mbt configuration aimed at treating time dependent compressible flows with moving impermeable boundary surfaces a strang type operator splitting is employed for the integration of the conservation laws the boundary motion is also split into cartesian components that are taken with the respective operator split integration phases the conservation laws in boundary cells are integrated by a finite volume scheme that accounts for changing mesh geometry cell volume and side area through which mass momentum and energy fluxes are taken the central feature of the scheme is the algorithms for evaluating the changing mesh geometry at boundary cells these algorithms are based on a decomposition rule for evaluating polygon intersection area which has been used extensively for rezoning in hydrocodes over the past decades the decomposition rule is combined with the cartesian splitting of the boundary motion producing algorithms for the integration of conservation laws in boundary cells that are both consistent and simple consistency is taken to mean that the scheme produces an exact solution in the case of a uniform motion of the fluid and the boundary common velocity an illustrative example of shock lifting of a light weight cylinder is presented

a versatile sharp interface immersed boundary method for incompressible flows with complex boundaries a sharp interface immersed boundary method for simulating incompressible viscous flow past three dimensional immersed bodies is described the method employs a multi dimensional ghost cell methodology to satisfy the boundary conditions on the immersed boundary and the method is designed to handle highly complex three dimensional stationary moving and or deforming bodies the complex immersed surfaces are represented by grids consisting of unstructured triangular elements while the flow is computed on non uniform cartesian grids the paper describes the salient features of the methodology with special emphasis on the immersed boundary treatment for stationary and moving boundaries simulations of a number of canonical two and three dimensional flows are used to verify the accuracy and fidelity of the solver over a range of reynolds numbers flow past suddenly accelerated bodies are used to validate the solver for moving boundary problems finally two cases inspired from biology with highly complex three dimensional bodies are simulated in order to demonstrate the versatility of the method

the filippov characteristic flow for the aggregation equation with mildly singular potentials existence and uniqueness of global in time measure solution for the multidimensional aggregation equation is analyzed such a system can be written as a continuity equation with a velocity eld computed through a self consistent interaction potential in carrillo et al duke math j 2011 16 a well posedness theory based on the geometric approach of gradient ows in measure metric spaces has been developed for mildly singular potentials at the origin under the basic assumption of being convex we propose here an alternative method using classical tools from pdes we show the existence of a characteristic ow based on filippov s theory of discontinuous dynamical systems such that the weak measure solution is the pushforward measure with this ow uniqueness is obtained thanks to a contraction argument in transport distances using the convexity of the potential moreover we show the equivalence of this solution with the gradient ow solution finally we show the convergence of a numerical scheme for general measure solutions in this framework allowing for the simulation of solutions for initial smooth densities after their rst blow up time in l p norms

continuum modeling of the equilibrium and stability of animal flocks abstract groups of animals often tend to arrange themselves in flocks that have characteristic spatial attributes and temporal dynamics using a dynamic continuum model for a flock of individuals we find equilibria of finite spatial extent where the density goes continuously to zero at a well defined flock edge and we discuss conditions on the model that allow for such solutions we also demonstrate conditions under which as the flock size increases the interior density in our equilibria tends to an approximately uniform value motivated by observations of starling flocks that are relatively thin in a direction transverse to the direction of flight we investigate the stability of infinite planar sheet flock equilibria we find that long wavelength perturbations along the sheet are unstable for the class of models that we investigate this has the conjectured consequence that sheet like flocks of arbitrarily large transverse extent relative to their thickness do not occur however we also show that our model admits approximately sheet like pancake shaped three dimensional ellipsoidal equilibria with definite aspect ratios transverse length scale to flock thickness determined by anisotropic perceptual response characteristics of the flocking individuals and we argue that these pancake like equilibria are stable to the previously mentioned sheet instability

a representation of curved boundaries for the solution of the navier stokes equations on a staggered three dimensional cartesian grid a method is presented for representing curved boundaries for the solution of the navier stokes equations on a non uniform staggered three dimensional cartesian grid the approach involves truncating the cartesian cells at the boundary surface to create new cells which conform to the shape of the surface we discuss in some detail the problems unique to the development of a cut cell method on a staggered grid methods for calculating the fluxes through the boundary cell faces for representing pressure forces and for calculating the wall shear stress are derived and it is verified that the new scheme retains second order accuracy in space in addition a novel cell linking method is developed which overcomes problems associated with the creation of small cells while avoiding the complexities involved with other cell merging approaches techniques are presented for generating the geometric information required for the scheme based on the representation of the boundaries as quadric surfaces the new method is tested for flow through a channel placed oblique to the grid and flow past a cylinder at re 40 and is shown to give significant improvement over a staircase boundary formulation finally it is used to calculate unsteady flow past a hemispheric protuberance on a plate at a reynolds number of 800 good agreement is obtained with experimental results for this flow

delay times of sequential procedures for multiple time series regression models we consider a multiple regression model in which the explanatory variables are specified by time series to sequentially test for the stability of the regression parameters in time we introduce a detector which is based on the first excess time of a cusum type statistic over a suitably constructed threshold function the aim of this paper is to study the delay time associated with this detector as our main result we derive the limit distribution of the delay time and provide thereby a theory that extends the benchmark average run length concept utilized in most of the sequential monitoring literature to highlight the applicability of the limit results in finite samples we present a monte carlo simulation study and an application to macroeconomic data

asymptotic nonequivalence of garch models and diffusions this paper investigates the statistical relationship of the garch model and its diffusion limit regarding the two types of models as two statistical experiments formed by discrete observations from the models we study their asymptotic equivalence in terms of le cam s deficiency distance to our surprise we are able to show that the garch model and its diffusion limit are asymptotically equivalent only under deterministic volatility with stochastic volatility due to the difference between the structure with respect to noise propagation in their conditional variances their likelihood processes asymptotically behave quite differently and thus they are not asymptotically equivalent this stochastic nonequivalence discredits a general belief that the two types of models are asymptotically equivalent in all respects and warns against the common financial practice that applies statistical inferences derived under the garch model to its diffusion limit

gaslike model of social motility we propose a model to represent the motility of social elements the model is completely deterministic possesses a small number of parameters and exhibits a series of properties that are reminiscent of the behavior of comunities in social ecological competition these are i similar individuals attract each other ii individuals can form stable groups iii a group of similar individuals breaks into subgroups if it reaches a critical size iv interaction between groups can modify the distribution of the elements as a result of fusion fission or pursuit v individuals can change their internal state by interaction with their neighbors the simplicity of the model and its richness of emergent behaviors such as for example pursuit between groups make it a useful toy model to explore a diversity of situations by changing the rule by which the internal state of individuals is modified by the interactions with the environment

cardinality estimation of support of the global minimizer for the interaction energy with mildly repulsive potentials abstract in this work we give a cardinality estimation of the support of any global minimizer for the interaction energy with mildly repulsive potentials for any global minimizer the fact that supp consists of finitely many points was proved in carrillo et al 2017 moreover the authors found a cardinality estimation of the support of on r 1 in order to advance some improvement of this result we give an upper estimate of card supp in arbitrary finite dimension

an immersed boundary finite volume method for simulations of flow in complex geometries abstract a new immersed boundary method for simulating flows over or inside complex geometries is developed by introducing a mass source sink as well as a momentum forcing the present method is based on a finite volume approach on a staggered mesh together with a fractional step method both momentum forcing and mass source are applied on the body surface or inside the body to satisfy the no slip boundary condition on the immersed boundary and also to satisfy the continuity for the cell containing the immersed boundary in the immersed boundary method the choice of an accurate interpolation scheme satisfying the no slip condition on the immersed boundary is important because the grid lines generally do not coincide with the immersed boundary therefore a stable second order interpolation scheme for evaluating the momentum forcing on the body surface or inside the body is proposed three different flow problems decaying vortices and flows over a cylinder and a sphere are simulated using the immersed boundary method proposed in this study and the results agree very well with previous numerical and experimental results verifying the accuracy of the present method

new tools for characterizing swarming systems a comparison of minimal models abstract we compare three simple models that reproduce qualitatively the emergent swarming behavior of bird flocks fish schools and other groups of self propelled agents by using a new set of diagnosis tools related to the agents spatial distribution two of these correspond in fact to different implementations of the same model which had been previously confused in the literature all models appear to undergo a very similar order to disorder phase transition as the noise level is increased if we only compare the standard order parameter which measures the degree of agent alignment when considering our novel quantities however their properties are clearly distinguished unveiling previously unreported qualitative characteristics that help determine which model best captures the main features of realistic swarms additionally we analyze the agent clustering in space finding that the distribution of cluster sizes is typically exponential at high noise and approaches a power law as the noise level is reduced this trend is sometimes reversed at noise levels close to the phase transition suggesting a non trivial critical behavior that could be verified experimentally finally we study a bi stable regime that develops under certain conditions in large systems by computing the probability distributions of our new quantities we distinguish the properties of each of the coexisting metastable states our study suggests new experimental analyses that could be carried out to characterize real biological swarms

computation of solid liquid phase fronts in the sharp interface limit on fixed grids a finite difference formulation is applied to track solid liquid boundaries on a fixed underlying grid the interface is not of finite thickness but is treated as a discontinuity and is explicitly tracked the imposition of boundary conditions exactly on a sharp interface that passes through the cartesian grid is performed using simple stencil readjustments in the vicinity of the interface attention is paid to formulating difference schemes that are globally second order accurate in x and t error analysis and grid refinement studies are performed for test problems involving the diffusion and convection diffusion equations and for stable solidification problems issues concerned with stability and change of phase of grid points in the evolution of solid liquid phase fronts are also addressed it is demonstrated that the field calculation is second order accurate while the position of the phase front is calculated to first order accuracy furthermore the accuracy estimates hold for the cases where there is a property jump across the interface unstable solidification phenomena are simulated and an attempt is made to compare results with previously published work the results indicate the need to begin an effort to benchmark computations of instability phenomena

self similar blowup solutions to an aggregation equation in r n we present numerical simulations of radially symmetric finite time blowup for the aggregation equation ut u k u where the kernel k x x the dynamics of the blowup exhibits self similar behavior in which zero mass concentrates at the core at the blowup time computations are performed in r n for n between 2 and 10 using a method based on characteristics in all cases studied the self similarity exhibits second kind anomalous scaling

immersed boundary method for flow around an arbitrarily moving body for the simulation of flow around an arbitrarily moving body an immersed boundary method is developed in a non inertial reference frame that is fixed to the body the navier stokes equation is formulated in a conservative form such that the force terms due to the rotation and the translational and rotational accelerations are included in the nonlinear term in order to satisfy the no slip condition on the body surface momentum forcing and mass source sink are applied on the body surface or inside the body the numerical method is based on a finite volume approach on a staggered mesh together with a fractional step method the present numerical method is applied to both the forced motion and fluid structure interaction problems in the latter we solve fully coupled navier stokes and dynamic equations for the moving body without introducing any iteration four different flow problems are tested and the results are in excellent agreements with previous numerical and experimental ones

generalized autoregressive conditional heteroskedasticity a natural generalization of the arch autoregressive conditional heteroskedastic process introduced in engle 1982 to allow for past conditional variances in the current conditional variance equation is proposed stationarity conditions and autocorrelation structure for this new class of parametric models are derived maximum likelihood estimation and testing are also considered finally an empirical example relating to the uncertainty of the inflation rate is presented

stability analysis of swarms in this note we specify an individual based continuous time model for swarm aggregation in n dimensional space and study its stability properties we show that the individuals autonomous agents or biological creatures will form a cohesive swarm in a finite time moreover we obtain an explicit bound on the swarm size which depends only on the parameters of the swarm model

global regularity for the fractional euler alignment system we study a pressureless euler system with a non linear density dependent alignment term originating in the cucker smale swarming models the alignment term is dissipative in the sense that it tends to equilibrate the velocities its density dependence is natural the alignment rate increases in the areas of high density due to species discomfort the diffusive term has the order of a fractional laplacian partial xx alpha 2 alpha in 0 1 the corresponding burgers equation with a linear dissipation of this type develops shocks in a finite time we show that the alignment nonlinearity enhances the dissipation and the solutions are globally regular for all alpha in 0 1 to the best of our knowledge this is the first example of such regularization due to the non local nonlinear modulation of dissipation

a semiparametric garch model for foreign exchange volatility abstract a semiparametric extension of the gjr model glosten et al 1993 journal of finance 48 1779 1801 is proposed for the volatility of foreign exchange returns under reasonable assumptions asymptotic normal distributions are established for the estimators of the model corroborated by simulation results when applied to the deutsche mark us dollar and the deutsche mark british pound daily returns data the semiparametric volatility model outperforms the gjr model as well as the more commonly used garch 1 1 model in terms of goodness of fit and forecasting by correcting overgrowth in volatility

an accurate moving boundary formulation in cut cell methods a cut cell method for cartesian meshes to simulate viscous compressible flows with moving boundaries is presented we focus on eliminating unphysical oscillations occurring in cartesian grid methods extended to moving boundary problems in these methods cells either lie completely in the fluid or solid region or are intersected by the boundary for the latter cells the time dependent volume fraction lying in the fluid region can be so small that explicit time integration schemes become unstable and a special treatment of these cells is necessary when the boundary moves a fluid cell may become a cut cell or a solid cell may become a small cell at the next time level this causes an abrupt change in the discretization operator and a suddenly modified truncation error of the numerical scheme this temporally discontinuous alteration is shown to act like an unphysical source term which deteriorates the numerical solution i e it generates unphysical oscillations in the hydrodynamic forces exerted on the moving boundary we develop an accurate moving boundary formulation based on the varying discretization operators yielding a cut cell method which avoids these discontinuities results for canonical two and three dimensional test cases evidence the accuracy and robustness of the newly developed scheme

combined immersed boundary finite difference methods for three dimensional complex flow simulations a second order accurate highly efficient method is developed for simulating unsteady three dimensional incompressible flows in complex geometries this is achieved by using boundary body forces that allow the imposition of the boundary conditions on a given surface not coinciding with the computational grid the governing equations therefore can be discretized and solved on a regular mesh thus retaining the advantages and the efficiency of the standard solution procedures two different forcings are tested showing that while the quality of the results is essentially the same in both cases the efficiency of the calculation strongly depends on the particular expression a major issue is the interpolation of the forcing over the grid that determines the accuracy of the scheme this ranges from zeroth order for the most commonly used interpolations up to second order for an ad hoc velocity interpolation the present scheme has been used to simulate several flows whose results have been validated by experiments and other results available in the literature finally in the last example we show the flow inside an ic piston cylinder assembly at high reynolds number to our knowledge this is the first example in which the immersed boundary technique is applied to a full three dimensional complex flow with moving boundaries and with a reynolds number high enough to require a subgrid scale turbulence model

an efficient conservative cut cell method for rigid bodies interacting with viscous compressible flows a cartesian cut cell method for viscous flows interacting with freely moving boundaries is presented the method enables a sharp resolution of the embedded boundaries and strictly conserves mass momentum and energy a new explicit runge kutta scheme pc rk is introduced by which the overall computational time is reduced by a factor of up to 2 5 the new scheme is a predictor corrector type reformulation of a popular class of runge kutta methods which substantially reduces the computational effort for tracking the moving boundaries and subsequently reinitializing the solver impairing neither stability nor accuracy the structural motion is computed by an implicit scheme with good stability properties due to a strong coupling strategy and the conservative discretization of the flow solver at the material interfaces a new formulation for the treatment of small cut cells is proposed with high accuracy and robustness for arbitrary geometries based on a weighted taylor series approach solved via singular value decomposition the efficiency and the accuracy of the new method are demonstrated for several three dimensional cases of laminar and turbulent particulate flow it is shown that the new method remains fully conservative even for large displacements of the boundaries leading to a fast convergence of the fluid solid coupling while spurious force oscillations inherent to this class of methods are effectively suppressed the results substantiate the good stability and accuracy properties of the scheme even on relatively coarse meshes an efficient cartesian cut cell method for freely moving solid boundaries is presented the new predictor corrector runge kutta scheme pc rk enables simulation speedups of up to 2 5 mass momentum and energy are strictly conserved at the sharply resolved solid fluid interfaces the method exhibits very good stability and accuracy properties even for large displacements of the boundaries spurious force oscillations inherent to this class of methods are suppressed

emergent dynamics of winfree oscillators on locally coupled networks abstract the winfree model is the first mathematical model for synchronization of weakly coupled oscillators compared to the well known kuramoto model the winfree model does not conserve the total phase this leads to rich dynamic features compared to those produced by other phase models in this paper we study the emergent dynamics of the winfree model on a locally coupled static network a randomly chosen phase configuration undergoes several dynamic phase transitions such as incoherence partial locking complete locking partial oscillator death and complete oscillator death as the coupling strength increases we provide several rigorous analytical results on the emergence of these dynamic features we also provide several numerical simulations and compare their results to the analytical results

compactness for nonlinear continuity equations abstract we prove compactness and hence existence for solutions to a class of nonlinear transport equations the corresponding models combine the features of linear transport equations and scalar conservation laws we introduce a new method which gives quantitative compactness estimates compatible with both frameworks

cucker smale flocking under hierarchical leadership a mathematical theory on flocking serves the foundation for several ubiquitous multiagent phenomena in biology ecology sensor networks and economics as well as social behavior like language emergence and evolution directly inspired by the recent fundamental works of cucker and smale on the construction and analysis of a generic flocking model we study the emergent behavior of cucker smale flocking under hierarchical leadership the rates of convergence towards asymptotically coherent group patterns in different scenarios are established the consistent convergence towards coherent patterns may well reveal the advantages and necessities of having leaders and leadership in a complex biological technological economic or social system with sufficient intelligence

gibbs sampler based coordination of autonomous swarms in this paper a novel gibbs sampler based algorithm is proposed for coordination of autonomous swarms the swarm is modeled as a markov random field mrf on a graph with a time varying neighborhood system determined by local interaction links the gibbs potential is designed to reflect global objectives and constraints it is established that with primarily local sensing communications the swarm configuration converges to the global minimizer s of the potential function the impact of the gibbs potential on the convergence speed is investigated finally a hybrid algorithm is developed to improve the efficiency of the stochastic scheme by integrating the gibbs sampler based method with the deterministic gradient flow method simulation results are presented to illustrate the proposed approach and verify the analyses

finite time blow up of solutions of an aggregation equation in rn we consider the aggregation equation u t nabla cdot u nabla k u 0 in rn n 2 where k is a rotationally symmetric nonnegative decaying kernel with a lipschitz point at the origin e g k x e x we prove finite time blow up of solutions from specific smooth initial data for which the problem is known to have short time existence of smooth solutions

a conservative immersed interface method for large eddy simulation of incompressible flows we propose a conservative second order accurate immersed interface method for representing incompressible fluid flows over complex three dimensional solid obstacles on a staggered cartesian grid the method is based on a finite volume discretization of the incompressible navier stokes equations which is modified locally in cells that are cut by the interface in such a way that accuracy and conservativity are maintained a level set technique is used for description and tracking of the interface geometry so that an extension of the method to moving boundaries and flexible walls is straightforward numerical stability is ensured for small cells by a conservative mixing procedure discrete conservation and sharp representation of the fluid solid interface render the method particularly suitable for large eddy simulations of high reynolds number flows accuracy second order grid convergence and robustness of the method is demonstrated for several test cases inclined channel flow at re 20 flow over a square cylinder at re 100 flow over a circular cylinder at re 40 re 100 and re 3900 as well as turbulent channel flow with periodic constrictions at re 10 595

singular patterns for an aggregation model with a confining potential abstract we consider the aggregation equation with an attractive repulsive force law recent studies kolokolnikov et al 2011 22 von brecht et al 2012 23 balague et al 2013 15 have demonstrated that this system exhibits a very rich solution structure including steady states consisting of rings spots annuli n fold symmetries soccer ball patterns etc we show that many of these patterns can be understood as singular perturbations off lower dimensional equilibrium states for example an annulus is a bifurcation from a ring soccer ball patterns bifurcate off solutions that consist of delta point concentrations we apply asymptotic methods to classify the form and stability of many of these patterns to characterize spot solutions a class of semi linear aggregation problems is derived where the repulsion is described by a nonlinear term and the attraction is linear but non symmetric for a special class of perturbations that consists of a newtonian repulsion the spot shape is shown to be an ellipse whose precise dimensions are determined via a complex variable method for annular shapes their width and radial density profile are described using perturbation techniques

flocking estimates for the cucker smale model with time lag and hierarchical leadership abstract we analyze the cucker smale model under hierarchical leadership in presence of a time delay by using a lyapunov functional approach and some induction arguments we will prove convergence to consensus for every positive delay we also prove a flocking estimate in the case of a free will leader these results seem to point out the advantage of a hierarchical structure in order to contrast time delay effects that frequently appear in real situations

on the order of accuracy of the immersed boundary method higher order convergence rates for sufficiently smooth problems the immersed boundary method is both a mathematical formulation and a numerical scheme for problems involving the interaction of a viscous incompressible fluid and a visco elastic structure in m c lai simulations of the flow past an array of circular cylinders as a test of the immersed boundary method ph d thesis courant institute of mathematical sciences new york university 1998 m c lai c s peskin an immersed boundary method with formal second order accuracy and reduced numerical viscosity j comput phys 160 2000 705 719 lai and peskin introduced a formally second order accurate immersed boundary method but the convergence properties of their algorithm have only been examined computationally for problems with nonsmooth solutions consequently in practice only first order convergence rates have been observed in the present work we describe a new formally second order accurate immersed boundary method and demonstrate its performance for a prototypical fluid structure interaction problem involving an immersed viscoelastic shell of finite thickness studied over a broad range of reynolds numbers we consider two sets of material properties for the viscoelastic structure including a case where the material properties of the coupled system are discontinuous at the fluid structure interface for both sets of material properties the true solutions appear to possess sufficient smoothness for the method to converge at a second order rate for fully resolved computations

from discrete cucker smale model to continuous cucker smale model in a temperature field we study emergent dynamics of the discrete and continuous cucker smale c s models in a self consistent temperature field and provide refined monocluster flocking estimates more precisely we present a sufficient framework leading to the monocluster flocking toward the constant common temperature and velocity exponentially fast this improves the authors earlier work where existence of common temperature and velocity are not identified our proposed sufficient framework is formulated in terms of the initial data and system parameters using the refined flocking we also provide a uniform in time convergence from the discrete model to the continuous model under some suitable compatibility condition on the asymptotic velocities for the discrete and continuous model as the time step tends to zero we also provide several numerical examples and compare them with analytical results we study emergent dynamics of the discrete and continuous cucker smale c s models in a self consistent temperature field and provide refined monocluster flocking estimates more precisely we present a sufficient framework leading to the monocluster flocking toward the constant common temperature and velocity exponentially fast this improves the authors earlier work where existence of common temperature and velocity are not identified our proposed sufficient framework is formulated in terms of the initial data and system parameters using the refined flocking we also provide a uniform in time convergence from the discrete model to the continuous model under some suitable compatibility condition on the asymptotic velocities for the discrete and continuous model as the time step tends to zero we also provide several numerical examples and compare them with analytical results

contour dynamics for the euler equations in two dimensions we present a contour dynamics algorithm for the euler equations of fluid dynamics in two dimensions this is applied to regions of piecewise constant vorticity withinfinite area vortexregions favrs essentially this reduces the dimensionality by one and we are treating the interaction of closed polygonal contours whose nodes are advected by the total fluid motion computed self consistently a leapfrog centered scheme is used for temporal advancement computer simulation results are given for two and four like signed interacting favrs in all cases wavelike surface deformations are observed if the distance between favrs is comparable to their extent diameter these surface deformations are large they play an essential role in the observed coalescence of favrs

an adaptive version of the immersed boundary method a computational setting for the immersed boundary method employing an adaptive mesh refinement is presented enhanced accuracy for the method is attained locally by covering an immersed boundary vicinity with a sequence of nested progressively finer rectangular grid patches which dynamically follow the immersed boundary motion the set of equations describing the interaction between a non stationary viscous incompressible fluid and an immersed elastic boundary is solved by coupling a projection method specially designed for locally refined meshes to an implicit formulation of the immersed boundary method the main contributions of this work concern the formulation and the implementation of a multilevel self adaptive version of the immersed boundary method on locally refined meshes this approach is tested for a particular two dimensional model problem for which no significant difference is found between the solutions obtained on a mesh refined locally around the immersed boundary and on the associated uniform mesh built with the resolution of the finest level

an embedded boundary formulation for large eddy simulation of turbulent flows interacting with moving boundaries a non boundary conforming formulation for simulating complex turbulent flows with dynamically moving boundaries on fixed cartesian grids is proposed the underlying finite difference solver for the filtered incompressible navier stokes equations is based on a second order fractional step method on a staggered grid to satisfy the boundary conditions on an arbitrary immersed interface the velocity field at the grid points near the interface is reconstructed using momentum forcing without smearing the sharp interface the concept of field extension is also introduced to treat the points emerging from a moving solid body to the fluid laminar flow cases and large eddy simulations les are presented to demonstrate the formal accuracy and range of applicability of the method in particular simulations of laminar flow induced by the harmonic in line oscillation of a circular cylinder in quiescent fluid and from a transversely oscillating cylinder in a free stream are presented and compared to reference simulations and experiments les of turbulent flow over a traveling wavy wall and transitional flow through a bileaflet prosthetic heart valve are also shown all results are in very good agreement with reference results in the literature

equilibria and steering laws for planar formations abstract this paper presents a lie group setting for the problem of control of formations as a natural outcome of the analysis of a planar two vehicle formation control law the vehicle trajectories are described using the planar frenet serret equations of motion which capture the evolution of both the vehicle position and orientation for unit speed motion subject to curvature steering control the set of all possible relative equilibria for arbitrary g invariant curvature controls is described where g se 2 is a symmetry group for the control law and a global convergence result for the two vehicle control law is proved an n vehicle generalization of the two vehicle control law is also presented and the corresponding relative equilibria for the n vehicle problem are characterized work is on going to discover stability and convergence results for the n vehicle problem

enhancing consensus in weighted networks with coupling time delay abstract we study the consensus problem in a weighted and directed network composed of self propelled agents the weight which quantifies the relationship between the agents is adjusted according to their state incoherence to suppress the heterogeneity in state the proposed consensus protocol enhances the convergence efficiency of consensus greatly and has better performance than that of two other protocols the convergence efficiency can be further improved by adjusting the parameter of this protocol furthermore the robustness of the system against coupling time delay is significantly increased unlike most consensus acceleration methods no topological information is needed in our model networks with different stuctures are investigated to demonstrate the generality of the method

interaction ruling animal collective behavior depends on topological rather than metric distance evidence from a field study numerical models indicate that collective animal behavior may emerge from simple local rules of interaction among the individuals however very little is known about the nature of such interaction so that models and theories mostly rely on aprioristic assumptions by reconstructing the three dimensional positions of individual birds in airborne flocks of a few thousand members we show that the interaction does not depend on the metric distance as most current models and theories assume but rather on the topological distance in fact we discovered that each bird interacts on average with a fixed number of neighbors six to seven rather than with all neighbors within a fixed metric distance we argue that a topological interaction is indispensable to maintain a flock s cohesion against the large density changes caused by external perturbations typically predation we support this hypothesis by numerical simulations showing that a topological interaction grants significantly higher cohesion of the aggregation compared with a standard metric one

reconsidering the continuous time limit of the garch 1 1 process in this note we reconsider the continuous time limit of the garch 1 1 process let k and p2 denote respectively the cumulative returns and the volatility processes we consider the continuous time approximation of the couple k p2 we show that by choosing di erent parameterizations as a function of the discrete interval h we can obtain either a degenerate or a non degenerate di usion limit we then show that garch 1 1 processes can be obtained as euler approximations of degenerate di usions while any euler approximation of a non degenerate di usion is a stochastic volatility process 2000 elsevier science s a all rights reserved

global in time weak measure solutions and finite time aggregation for nonlocal interaction equations in this paper we provide a well posedness theory for weak measure solutions of the cauchy problem for a family of nonlocal interaction equations these equations are continuum models for interacting particle systems with attractive repulsive pairwise interaction potentials the main phenomenon of interest is that even with smooth initial data the solutions can concentrate mass in finite time we develop an existence theory that enables one to go beyond the blow up time in classical norms and allows for solutions to form atomic parts of the measure in finite time the weak measure solutions are shown to be unique and exist globally in time moreover in the case of sufficiently attractive potentials we show the finite time total collapse of the solution onto a single point for compactly supported initial measures our approach is based on the theory of gradient flows in the space of probability measures endowed with the wasserstein metric in addition to classical tools we exploit the stability of the flow with respect to the transportation distance to greatly simplify many problems by reducing them to questions about particle approximations

a generalized birkhoff rott equation for two dimensional active scalar problems in this paper we derive evolution equations for the two dimensional active scalar problem when the solution is supported on one dimensional curves these equations are a generalization of the birkhoff rott equation when vorticity is the active scalar the formulation is lagrangian and it is valid for nonlocal kernels bf k that may include both a gradient and an incompressible term we develop a numerical method for implementing the model which achieves second order convergence in space and fourth order in time we verify the model by simulating classic active scalar problems such as the vortex sheet problem in the case of inviscid incompressible flow and the collapse of delta ring solutions in the case of pure aggregation finding excellent agreement we then study two examples with kernels of mixed type i e kernels that contain both incompressible and gradient flows the first example is a vortex density model which arises in superfluids we analyze the effect of the added gradient component

swarming patterns in a two dimensional kinematic model for biological groups we construct a continuum model for the motion of biological organisms experiencing social interactions and study its pattern forming behavior the model takes the form of a conservation law in two spatial dimensions the social interactions are modeled in the velocity term which is nonlocal in the population density and includes a parameter that controls the interaction length scale the dynamics of the resulting partial integrodifferential equation may be uniquely decomposed into incompressible motion and potential motion for the purely incompressible case the model resembles one for fluid dynamical vortex patches there exist solutions which have constant population density and compact support for all time numerical simulations produce rotating structures which have circular cores and spiral arms and are reminiscent of naturally observed phenomena such as ant mills the sign of the social interaction term determines the direction of the rotation and the interaction length scale affects the degree o

stationarity of garch processes and of some nonnegative time series abstract two time series models are considered garch processes and generalized multivariate autoregressive equations x n 1 a n 1 x n b n 1 with nonnegative i i d coefficients in each case a necessary and sufficient condition ensuring the existence of a strictly stationary solution is given

models of collective cell motion for cell populations with different aspect ratio diffusion proliferation and travelling waves continuum partial differential equation models are often used to describe the collective motion of cell populations with various types of motility represented by the choice of diffusion coefficient and cell proliferation captured by the source terms previously the choice of diffusion coefficient has been largely arbitrary with the decision to choose a particular linear or nonlinear form generally based on calibration arguments rather than making any physical connection with the underlying individual level properties of the cell motility mechanism in this work we provide a new link between individual level models which account for important cell properties such as varying cell shape and volume exclusion and population level partial differential equation models we work in an exclusion process framework considering aligned elongated cells that may occupy more than one lattice site in order to represent populations of agents with different sizes three different idealisations of the individual level mechanism are proposed and these are connected to three different partial differential equations each with a different diffusion coefficient one linear one nonlinear and degenerate and one nonlinear and nondegenerate we test the ability of these three models to predict the population level response of a cell spreading problem for both proliferative and nonproliferative cases we also explore the potential of our models to predict long time travelling wave invasion rates and extend our results to two dimensional spreading and invasion our results show that each model can accurately predict density data for nonproliferative systems but that only one does so for proliferative systems hence great care must be taken to predict density data with varying cell shape