spectral lagrangian methods for collisional models of non equilibrium statistical states we propose a new spectral lagrangian based deterministic solver for the non linear boltzmann transport equation bte in d dimensions for variable hard sphere vhs collision kernels with conservative or non conservative binary interactions the method is based on symmetries of the fourier transform of the collision integral where the complexity in its computation is reduced to a separate integral over the unit sphere s d 1 the conservation of moments is enforced by lagrangian constraints the resulting scheme implemented in free space is very versatile and adjusts in a very simple manner to several cases that involve energy dissipation due to local micro reversibility inelastic interactions or elastic models of slowing down process our simulations are benchmarked with available exact self similar solutions exact moment equations and analytical estimates for the homogeneous boltzmann equation both for elastic and inelastic vhs interactions benchmarking of the simulations involves the selection of a time self similar rescaling of the numerical distribution function which is performed using the continuous spectrum of the equation for maxwell molecules as studied first in bobylev et al a v bobylev c cercignani g toscani proof of an asymptotic property of self similar solutions of the boltzmann equation for granular materials journal of statistical physics 111 2003 403 417 and generalized to a wide range of related models in bobylev et al a v bobylev c cercignani i m gamba on the self similar asymptotics for generalized non linear kinetic maxwell models communication in mathematical physics in press url the method also produces accurate results in the case of inelastic diffusive boltzmann equations for hard spheres inelastic collisions under thermal bath where overpopulated non gaussian exponential tails have been conjectured in computations by stochastic methods t v noije m ernst velocity distributions in homogeneously cooling and heated granular fluids granular matter 1 57 1998 m h ernst r brito scaling solutions of inelastic boltzmann equations with over populated high energy tails journal of statistical physics 109 2002 407 432 s j moon m d shattuck j swift velocity distributions and correlations in homogeneously heated granular media physical review e 64 2001 031303 i m gamba s rjasanow w wagner direct simulation of the uniformly heated granular boltzmann equation mathematical and computer modelling 42 2005 683 700 and rigorously proven in gamba et al i m gamba v panferov c villani on the boltzmann equation for diffusively excited granular media communications in mathematical physics 246 2004 503 541 39 and a v bobylev i m gamba v panferov moment inequalities and high energy tails for boltzmann equations with inelastic interactions journal of statistical physics 116 2004 1651 1682

inner and outer bounds for the solution set of parametric linear systems consider linear systems involving affine linear dependencies on interval parameters presented is a free c xsc software implementing a generalized parametric fixed point iteration method for verified enclosure of the parametric solution set some specific features of the corresponding algorithm concerning sharp enclosure of the contracting matrix and inner approximation of the solution enclosure are discussed

stability of a saddle node bifurcation under numerical approximations in this paper we show that the solution flows generated by a one parameter family of ordinary differential equations are stable under their numerical approximations in a vicinity of a saddle node our result sharpens the one in 1 and the proof is adapted from the method of sotomayor in 2 3

analysis of spectral methods for the homogeneous boltzmann equation the development of accurate and fast algorithms for the boltzmann collision integral and their analysis represent a challenging problem in scientific computing and numerical analysis recently several works were devoted to the derivation of spectrally accurate schemes for the boltzmann equation but very few of them were concerned with the stability analysis of the method in particular there was no result of stability except when the method is modified in order to enforce the posivity preservation which destroys the spectral accuracy in this paper we propose a new method to study the stability of homogeneous boltzmann equations perturbed by smoothed balanced operators which do not preserve positivity of the distribution this method takes advantage of the spreading property of the collision together with estimates on regularity and entropy production as an application we prove stability and convergence of spectral methods for the boltzmann equation when the discretization parameter is large enough with explicit bound

rigorous sensitivity analysis for systems of linear and nonlinear equations methods are presented for performing a rigorous sensitivity analysis of numerical problems with independent noncorrelated data for general systems of linear and nonlinear equations the methods may serve for the following two purposes first to bound the dependency of the solution on changes in the input data in contrast to condition numbers a componentwise sensitivity analysis of the solution vector is performed second to estimate the true solution set for problems the input data of which are afflicted with tolerances the methods presented are very effective with the additional property that due to an automatic error control mechanism every computed result is guaranteed to be correct examples are given for linear systems demonstrating that the computed bounds are in general very sharp interesting comparisons to traditional condition numbers are given

spectral method for a kinetic swarming model in this paper we present the first numerical method for a kinetic description of the vicsek swarming model the kinetic model poses a unique challenge as there is a distribution dependent collision invariant to satisfy when computing the interaction term we use a spectral representation linked with a discrete constrained optimization to compute these interactions to test the numerical scheme we investigate the kinetic model at different scales and compare the solution with the microscopic and macroscopic descriptions of the vicsek model we observe that the kinetic model captures key features such as vortex formation and traveling waves

on the computation of all extensions of a p adic field of a given degree let k be a p adic field it is well known that k has only finitely many extensions of a given finite degree krasner has given formulae for the number of extensions of a given degree and discriminant following his work we present an algorithm for the computation of generating polynomials for all extensions k k of a given degree and discriminant

topological structure preserving numerical simulations of dynamical models this paper brings together two methods producing numerical solutions with a statement of their quality the nonstandard finite difference method and the method of validated computing it deals with the construction and the analysis of reliable numerical discretizations of dynamical systems by employing these two techniques an epidemiological model is used as a model example for their combined application

a shadowing lemma approach to global error analysis for initial value odes the authors show that for dynamical systems that possess a type of piecewise hyperbolicity in which there is no decrease in the number of stable modes the global error in a numerical approximation may be obtained as a reasonable magnification of the local error in particular under certain conditions the authors prove the existence of a trajectory on an infinite time interval of the given ordinary differential equation uniformly close to a given numerically computed orbit of the same differential equation by allowing for different initial conditions for finite time intervals a general result is proved for obtaining a posteriori bounds on the global error based on computable quantities and on finding and bounding the norm of a right inverse of a particular matrix two methods for finding and bounding estimating the norm of a right inverse are considered one method is based upon the choice of the pseudo or generalized inverse the other method is based upon solving multipoint boundary value problems bv

the behavior of finite element solutions of semilinear parabolic problems near stationary points the qualitative behavior of spatially semidiscrete finite element solutions of a semilinear parabolic problem near an unstable hyperbolic equilibrium overline u is studied it is shown that any continuous trajectory is approximated by an appropriate discrete trajectory and vice versa as long as they remain in a sufficiently small neighborhood of overline u error bounds of optimal order in the l 2 and h 1 norms hold uniformly over arbitrarily long time intervals in particular the local stable and unstable manifolds of the discrete problem converge to their continuous counterparts therefore the discretized dynamical system has the same qualitative behavior near overline u as the continuous system

on equivariant global epsilon constants for certain dihedral extensions we consider a conjecture of bley and burns which relates the epsilon constant of the equivariant artin l function of a galois extension of number fields to certain natural algebraic invariants for an odd prime number p we describe an algorithm which either proves the conjecture for all degree 2p dihedral extensions of the rational numbers or finds a counterexample we apply this to show the conjecture for all degree 6 dihedral extensions of q the correctness of the algorithm follows from a finiteness property of the conjecture which we prove in full generality

stable attracting sets in dynamical systems and in their one step discretizations we consider a dynamical system described by a system of ordinary differential equations which possesses a compact attracting set of arbitrary shape under the assumption of uniform asymptotic stability of in the sense of lyapunov we show that discretized versions of the dynamical system involving one step numerical methods have nearby attracting sets h which are also uniformly asymptotically stable our proof uses the properties of a lyapunov function which characterizes the stability of

on invariant closed curves for one step methods we show that a one step method as applied to a dynamical system with a hyperbolic periodic orbit exhibits an invariant closed curve for sufficiently small step size this invariant curve converges to the periodic orbit with the order of the method and it inherits the stability of the periodic orbit the dynamics of the one step method on the invariant curve can be described by the rotation number for which we derive an asymptotic expression our results complement those of 2 3 where one step methods were shown to create invariant curves if the dynamical system has a periodic orbit which is stable in either time direction or if the system undergoes a hopf bifurcation

on the discretization of a partial differential equation in the neighborhood of a periodic orbit this article is concerned with the comparison of the dynamic of a partial differential equation and its time discretization we restrict our attention to the neighborhood of a hyperbolic periodic orbit we show that the discretization possesses an invariant closed curve near the periodic orbit and that the trajectories of the semigroups defined by the partial differential equations and its approximation are close in a sense to be precised provided that different data are allowed this answers partly an open problem posed in 4 examples of application to dissipative partial equations are provided

structural stability of flows under numerics abstract in our previous paper siam j math anal 28 1997 381 388 we showed that the qualitative properties of a morse smale gradient like flow are preserved by its discretization mapping obtained via numerical methods in this paper we extend the result to flows which satisfy axiom a and the strong transversality condition we prove that if p 2 t is a c p 1 flow on a compact manifold satisfying axiom a and the strong transversality condition and n h is a numerical method of step size h and order p then for all sufficiently small h there are a homeomorphism h h and a continuous real valued function h on m such that h h h h h x x n h h h x and h h is o h p close to the identity map on m

a numerical method for the accurate solution of the fokker planck landau equation in the nonhomogeneous case abstract a new approach for the accurate numerical solution of the fokker planck landau fpl equation in the nonhomogeneous case is presented the method couples through a time splitting algorithm a finite volume scheme for the transport with a fast spectral solver for the efficient solution of the collision operator recently introduced the scheme allows the use of different grids in the velocity space for the transport and the collision phases the use of a suitable explicit runge kutta solver for the time integration of the collision phase permits avoid once of excessive small time steps induced by the stiffness of the diffusive collision operator numerical results for both space homogeneous and space nonhomogeneous situations show the efficiency and the accuracy of the present method

aspects of backward error analysis of numerical odes abstract in this paper some preliminary cases are studied to see how the idea of backward error analysis carries over to numerical solution of ordinary differential equations the interest here is in one step methods and infinite time intervals

two concepts for numerical periodic solutions of ode s in computing a numerical approximation for a hyperbolic periodic orbit of an ordinary differential equation one can look either for periodic points of the mapping corresponding the numerical method or for an invariant set of that mapping here the one step methods are considered it is shown that both of these concepts for numerical approximation of periodic solutions are meaningful in the sense that the corresponding sets exist and are c k circles provided the method is accurate enough also both of these are accurate to the order of the method further some relations of these concepts are studied

kinetic models for topological nearest neighbor interactions we consider systems of agents interacting through topological interactions these have been shown to play an important part in animal and human behavior precisely the system consists of a finite number of particles characterized by their positions and velocities at random times a randomly chosen particle the follower adopts the velocity of its closest neighbor the leader we study the limit of a system size going to infinity and under the assumption of propagation of chaos show that the limit kinetic equation is a non standard spatial diffusion equation for the particle distribution function we also study the case wherein the particles interact with their k closest neighbors and show that the corresponding kinetic equation is the same finally we prove that these models can be seen as a singular limit of the smooth rank based model previously studied in 10 the proofs are based on a combinatorial interpretation of the rank as well as some concentration of measure arguments

on the equivariant tamagawa number conjecture for tate motives and unconditional annihilation results dfg collaborative research center 701 spectral structures and topological methods in mathematics

discretization of semilinear differential equations with an exponential dichotomy abstract a structural stability result for one step discretizations of semilinear differential equations with an exponential dichotomy in its linear part is presented and interpreted as a shadowing result estimates are given in terms of the stepsize and of the order of the discretization method chosen

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

numerical solution of the boltzmann equation i spectrally accurate approximation of the collision operator in this paper we show that the use of spectral galerkin methods for the approximation of the boltzmann equation in the velocity space permits one to obtain spectrally accurate numerical solutions at a reduced computational cost we prove that the spectral algorithm preserves the total mass and approximates with infinite order accuracy momentum and energy consistency of the method is also proved and a stability result for a smoothed positive scheme is given we demonstrate that the fourier coefficients associated with the collision kernel of the equation have a very simple structure and in some cases can be computed explicitly numerical examples for homogeneous test problems in two and three dimensions confirm the advantages of the method

dynamically consistent nonstandard finite difference schemes for epidemiological models this work is the numerical analysis and computational companion of the paper by kamgang and sallet j c kamgang g sallet computation of threshold conditions for epidemiological models and global stability of the disease free equilibrium dfe mathematical biosciences 213 2008 1 12 where threshold conditions for epidemiological models and the global stability of the disease free equilibrium dfe are studied we establish a discrete counterpart of the main continuous result that guarantees the global asymptotic stability gas of the dfe for general epidemiological models then we design nonstandard finite difference nsfd schemes in which the metzler matrix structure of the continuous model is carefully incorporated and both mickens rules world scientific singapore 1994 on the denominator of the discrete derivative and the nonlocal approximation of nonlinear terms are used in an innovative way as a result of these strategies our nsfd schemes are proved to be dynamically consistent with the continuous model i e they replicate their basic features including the gas of the dfe the linear stability of the endemic equilibrium ee the positivity of the solutions the dissipativity of the system and its inherent conservation law the general analysis is made detailed for the mseir model for which the nsfd theta method is implemented with emphasis on the computational aspects such as its convergence or local truncation error numerical simulations that illustrate the theory are provided

equivariant epsilon constant conjectures for weakly ramified extensions we study the local epsilon constant conjecture as formulated by breuning in breuning j london math soc 70 2 289 306 2004 this conjecture fits into the general framework of the equivariant tamagawa number conjecture etnc and should be interpreted as a consequence of the expected compatibility of the etnc with the functional equation of artin l functions let k mathbb q p be unramified under some mild technical assumption we prove breuning s conjecture for weakly ramified abelian extensions n k with cyclic ramification group as a consequence of breuning s local global principle we obtain the validity of the global epsilon constant conjecture as formulated in bley and burns proc lond math soc 87 3 545 590 2003 and of chinburg s omega 2 conjecture as stated in chinburg ann math 121 2 351 376 1985 for certain infinite families f e of weakly and wildly ramified extensions of number fields

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

runge kutta methods for dissipative and gradient dynamical systems the numerical approximation of dissipative initial value problems by fixed time stepping runge kutta methods is considered and the asymptotic features of the numerical and exact solutions are compared a general class of ordinary differential equations for which dissipativity is induced through an inner product is studied throughout this class arises naturally in many finite dimensional applications such as the lorenz equations and also from the spatial discretization of a variety of partial differential equations arising in applied mathematics it is shown that the numerical solution defined by an algebraically stable method has an absorbing set and is hence dissipative for any fixed step size h 0 the numerical solution is shown to define a dynamical system on the absorbing set if h is sufficiently small and hence a global attractor a h exists upper semicontinuity of a h at h 0 is established which shows that for h small every point on the numerical attractor is close to a point on the true global attractor a under the additional assumption that the problem is globally lipschitz it is shown that if h is sufficiently small any method with positive weights defines a dissipative dynamical system on the whole space and upper semicontinuity of a h at h 0 is again established for gradient systems with globally lipschitz vector fields it is shown that any runge kutta method preserves the gradient structure for h sufficiently small for general dissipative gradient systems it is shown that algebraically stable methods preserve the gradient structure within the absorbing set for h sufficiently small convergence of the numerical attractor is studied and for a dissipative gradient system with hyperbolic equilibria lower semicontinuity at h 0 is established thus for such a system a h converges to a in the hausdorff metric as h 0

on structural stability of ordinary differential equations with respect to discretization methods on compact m dimensional discs morse smale differential systems having no periodic orbits are considered the main result is that they are correctly reproduced by one step discretization methods for methods of order p and stepsize h sufficiently small the time h map of the induced local flow and the h discretized system are joined by a conjugacy o h p near to the identity the paper fits well in the rapidly growing list of results stating that hyperbolic transversal structures are preserved by discretization the proof relies heavily on techniques elaborated by robbin 1971 in establishing his structural stability theorem on self diffeomorphisms of compact c infty manifolds

numerical dynamics of a nonstandard finite difference method for a class of delay differential equations abstract this paper proposes a nonstandard finite difference nsfd method for a class of delay differential equations we prove that for any step size h 1 m m z this numerical method could intrinsically preserve the qualitative behavior of the dynamical system including the local stability of equilibrium the existence and the direction of hopf bifurcation and the stability of bifurcating periodic solution finally to illustrate the analytic results the nsfd method is used for a model of the survival of red blood cells in animals

on the numerical approximation of phase portraits near stationary points we show that the phase portrait of a dynamical system near a stationary hyperbolic point is reproduced correctly by numerical methods such as one step methods or multi step methods satisfying a strong root condition this means that any continuous trajectory can be approximated by an appropriate discrete trajectory and vice versa to the correct order of convergence and uniformly on arbitrarily large time intervals in particular the stable and unstable manifolds of the discretization converge to their continuous counterparts

equivariant epsilon constants discriminants and etale cohomology let l k be a finite galois extension of number fields we formulate and study a conjectural equality between an element of the relative algebraic k group k 0 mathbb z mathrm gal l k mathbb r which is constructed from the equivariant artin epsilon constant of l k and a sum of structural invariants associated to l k the precise conjecture is motivated by the requirement that a special case of the equivariant refinement of the tamagawa number conjecture of bloch and kato as formulated by flach and the second named author should be compatible with the functional equation of the associated l function we show that more concretely our conjecture also suggests a completely systematic refinement of the central approach and results of classical galois module theory in particular the evidence for our conjecture that we present here already strongly refines many of the main results of galois module theory

on frohlich s conjecture for rings of integers of tame extensions where is the complex conjugate of z the constant w z is called the artin root number of in m l j martinet showed that om is not always a free zf module when n q and when f is a quaternion group of order 8 subsequently in f5 a frtihlich showed that the class of u in this special case is determined by the sign of the artin root number of the irreducible sympletic character of 2 we now return to the general case and let ci tif be the class group of locally free zf modules given a locally free 2 f module x we denote its class by x philippe cassou nogu6s defined a class t w in ci f which is defined solely in terms of the values of artin root numbers of sympletic characters this class will be defined in w after numerous computations by fr6hlich et al cf f6 c t3 for instance fr6hlich conjectured the result given below which we prove in this paper

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

rigorous chaos verification in discrete dynamical systems abstract in this paper we show how interval analysis can be used to calculate rigorously valid enclosures of transversal homoclinic points in discrete dynamical systems the existence of such points guarantees that a smale horseshoe is embedded in the mapping which therefore is chaotic we take special interest in the 2 dimensional case and describe a method to obtain a computer assisted proof of chaos for these systems numerical results are presented for the standard map