graphical models for marked point processes based on local independence summary a new class of graphical models capturing the dependence structure of events that occur in time is proposed the graphs represent so called local independences meaning that the intensities of certain types of events are independent of some but not necessarilly all events in the past this dynamic concept of independence is asymmetric similar to granger non causality so the corresponding local independence graphs differ considerably from classical graphical models hence a new notion of graph separation which is called separation is introduced and implications for the underlying model as well as for likelihood inference are explored benefits regarding facilitation of reasoning about and understanding of dynamic dependences as well as computational simplifications are discussed

a fourier integrator for the cubic nonlinear schrodinger equation with rough initial data standard numerical integrators suffer from an order reduction when applied to nonlinear schrodinger equations with low regularity initial data for example standard strang splitting requires the b

on the performance of exponential integrators for problems in magnetohydrodynamics exponential integrators have been introduced as an efficient alternative to explicit and implicit methods for integrating large stiff systems of differential equations over the past decades these methods have been studied theoretically and their performance was evaluated using a range of test problems while the results of these investigations showed that exponential integrators can provide significant computational savings the research on validating this hypothesis for large scale systems and understanding what classes of problems can particularly benefit from the use of the new techniques is in its initial stages resistive magnetohydrodynamic mhd modeling is widely used in studying large scale behavior of laboratory and astrophysical plasmas in many problems numerical solution of mhd equations is a challenging task due to the temporal stiffness of this system in the parameter regimes of interest in this paper we evaluate the performance of exponential integrators on large mhd problems and compare them to a state of the art implicit time integrator both the variable and constant time step exponential methods of epirk type are used to simulate magnetic reconnection and the kevin helmholtz instability in plasma performance of these methods which are part of the epic software package is compared to the variable time step variable order bdf scheme included in the cvode part of sundials library we study performance of the methods on parallel architectures and with respect to magnitudes of important parameters such as reynolds lundquist and prandtl numbers we find that the exponential integrators provide superior or equal performance in most circumstances and conclude that further development of exponential methods for mhd problems is warranted and can lead to significant computational advantages for large scale stiff systems of differential equations such as mhd

splitting methods for the nonlocal fowler equation we consider a nonlocal scalar conservation law proposed by andrew c fowler to describe the dynamics of dunes and we develop a numerical procedure based on splitting methods to approximate its solutions we begin by proving the convergence of the well known lie formula which is an approximation of the exact solution of order one in time we next use the split step fourier method to approximate the continuous problem using the fast fourier transform and the finite difference method our numerical experiments confirm the theoretical results

fourth order time stepping for stiff pdes a modification of the exponential time differencing fourth order runge kutta method for solving stiff nonlinear pdes is presented that solves the problem of numerical instability in the scheme as proposed by cox and matthews and generalizes the method to nondiagonal operators a comparison is made of the performance of this modified exponential time differencing etd scheme against the competing methods of implicit explicit differencing integrating factors time splitting and fornberg and driscoll s sliders for the kdv kuramoto sivashinsky burgers and allen cahn equations in one space dimension implementation of the method is illustrated by short matlab programs for two of the equations it is found that for these applications with fixed time steps the modified etd scheme is the best

on the benney lin and kawahara equations abstract we establish global well posedness for the initial value problem ivp associated to the so called benney lin equation this model is a korteweg de vries equation perturbed by dissipative and dispersive terms which appears in fluid dynamics we also study the limiting behaviour of solutions to this ivp when the parameters of the perturbed terms approach 0

a splitting approach for the magnetic schrodinger equation the schrodinger equation in the presence of an external electromagnetic field is an important problem in computational quantum mechanics it also provides a nice example of a differential equation whose flow can be split with benefit into three parts after presenting a splitting approach for three operators with two of them being unbounded we exemplarily prove first order convergence of lie splitting in this framework the result is then applied to the magnetic schrodinger equation which is split into its potential kinetic and advective parts the latter requires special treatment in order not to lose the conservation properties of the scheme we discuss several options numerical examples in one two and three space dimensions show that the method of characteristics coupled with a nonequispaced fast fourier transform nfft provides a fast and reliable technique for achieving mass conservation at the discrete level

an exponential type integrator for the kdv equation we introduce an exponential type time integrator for the kdv equation and prove its first order convergence in h 1 for initial data in h 3 furthermore we outline the generalization of the presented technique to a second order method

stochastic kinetic models dynamic independence modularity and graphs the dynamic properties and independence structure of stochastic kinetic models skms are analyzed an skm is a highly multivariate jump process used to model chemical reaction networks particularly those in biochemical and cellular systems we identify skm subprocesses with the corresponding counting processes and propose a directed cyclic graph the kinetic independence graph or kig that encodes the local independence structure of their conditional intensities given a partition a d b of the vertices the graphical separation a b d in the undirected kig has an intuitive chemical interpretation and implies that a is locally independent of b given a d it is proved that this separation also results in global independence of the internal histories of a and b conditional on a history of the jumps in d which under conditions we derive corresponds to the internal history of d the results enable mathematical definition of a modularization of an skm using its implied dynamics graphical decomposition methods are developed for the identification and efficient computation of nested modularizations application to an skm of the red blood cell advances understanding of this biochemical system

generalized integrating factor methods for stiff pdes the integrating factor if method for numerical integration of stiff nonlinear pdes has the disadvantage of producing large error coefficients when the linear term has large norm we propose a generalization of the if method and in particular construct multistep type methods with several orders of magnitude improved accuracy we also consider exponential time differencing etd methods and point out connections with a particular application of the commutator free lie group methods we present a new fourth order etdrk method with improved accuracy the methods considered are compared in several numerical examples

on the error propagation of semi lagrange and fourier methods for advection problems in this paper we study the error propagation of numerical schemes for the advection equation in the case where high precision is desired the numerical methods considered are based on the fast fourier transform polynomial interpolation semi lagrangian methods using a lagrange or spline interpolation and a discontinuous galerkin semi lagrangian approach which is conservative and has to store more than a single value per cell we demonstrate by carrying out numerical experiments that the worst case error estimates given in the literature provide a good explanation for the error propagation of the interpolation based semi lagrangian methods for the discontinuous galerkin semi lagrangian method however we find that the characteristic property of semi lagrangian error estimates namely the fact that the error increases proportionally to the number of time steps is not observed we provide an explanation for this behavior and conduct numerical simulations that corroborate the different qualitative features of the error in the two respective types of semi lagrangian methods the method based on the fast fourier transform is exact but due to round off errors susceptible to a linear increase of the error in the number of time steps we show how to modify the cooley tukey algorithm in order to obtain an error growth that is proportional to the square root of the number of time steps finally we show for a simple model that our conclusions hold true if the advection solver is used as part of a splitting scheme

viehweg s hyperbolicity conjecture is true over compact bases abstract we prove viehweg s hyperbolicity conjecture over compact bases and over bases with non uniruled compactification the most general case of the conjecture states that the base space of a maximal variation family of smooth projective manifolds with semi ample canonical sheaf is of log general type

a general dynamical statistical model with causal interpretation summary we develop a general dynamical model as a framework for causal interpretation we first state a criterion of local independence in terms of measurability of processes that are involved in the doob meyer decomposition of stochastic processes then we define direct and indirect influence we propose a definition of causal influence using the concepts of a physical system this framework makes it possible to link descriptive and explicative statistical models and encompasses quantitative processes and events one of the features of the paper is the clear distinction between the model for the system and the model for the observation we give a dynamical representation of a conventional joint model for human immunodeficiency virus load and cd4 cell counts we show its inadequacy to capture causal influences whereas in contrast known mechanisms of infection by the human immunodeficiency virus can be expressed directly through a system of differential equations

exponential time differencing for stiff systems we develop a class of numerical methods for stiff systems based on the method of exponential time differencing we describe schemes with second and higher order accuracy introduce new runge kutta versions of these schemes and extend the method to show how it may be applied to systems whose linear part is nondiagonal we test the method against other common schemes including integrating factor and linearly implicit methods and show how it is more accurate in a number of applications we apply the method to both dissipative and dispersive partial differential equations after illustrating its behavior using forced ordinary differential equations with stiff linear parts

viehweg s hyperbolicity conjecture for families with maximal variation we use the theory of hodge modules to construct viehweg zuo sheaves on base spaces of families with maximal variation and fibers of general type and more generally families whose geometric generic fiber has a good minimal model combining this with a result of campana p un we deduce viehweg s hyperbolicity conjecture in this context namely the fact that the base spaces of such families are of log general type this is approached as part of a general problem of identifying what spaces can support hodge theoretic objects with certain positivity properties

fourth order time stepping for kadomtsev petviashvili and davey stewartson equations purely dispersive partial differential equations such as the korteweg de vries equation the nonlinear schrodinger equation and higher dimensional generalizations thereof can have solutions which develop a zone of rapid modulated oscillations in the region where the corresponding dispersionless equations have shocks or blow up to numerically study such phenomena fourth order time stepping in combination with spectral methods is beneficial in resolving the steep gradients in the oscillatory region we compare the performance of several fourth order methods for the kadomtsev petviashvili and the davey stewartson equations two integrable equations in 2 1 dimensions these methods are exponential time differencing integrating factors time splitting implicit runge kutta and driscoll s composite runge kutta method the accuracy in the numerical conservation of integrals of motion is discussed

operator splitting methods for generalized korteweg de vries equations we apply the method of operator splitting on the generalized korteweg de vries kdv equation ut f u x uxxx 0 by solving the nonlinear conservation law ut f u x 0 and the linear dispersive equation ut uxxx 0 sequentially we prove that if the approximation obtained by operator splitting converges then the limit function is a weak solution of the generalized kdv equation convergence properties are analyzed numerically by studying the effect of combining different numerical methods for each of the simplified problems

on splitting methods for schrodinger poisson and cubic nonlinear schrodinger equations we give an error analysis of strang type splitting integrators for nonlinear schrodinger equations for schrodinger poisson equations with an h 4 regular solution a first order error bound in the h 1 norm is shown and used to derive a second order error bound in the l 2 norm for the cubic schrodinger equation with an h 4 regular solution first order convergence in the h 2 norm is used to obtain second order convergence in the l 2 norm basic tools in the error analysis are lie commutator bounds for estimating the local error and h m conditional stability for error propagation where m 1 for the schrodinger poisson system and m 2 for the cubic schrodinger equation

a rigorous derivation of the chemical master equation it is widely believed that the chemical master equation has no rigorous microphysical basis and hence no a priori claim to validity this view is challenged here through arguments purporting to show that the chemical master equation is exact for any gas phase chemical system that is kept well stirred and thermally equilibrated

a split step fourier discontinuous galerkin scheme for the kadomtsev petviashvili equation abstract in this paper we propose a method to solve the kadomtsev petviashvili equation based on splitting the linear part of the equation from the nonlinear part the linear part is treated using ffts while the nonlinear part is approximated using a semi lagrangian discontinuous galerkin approach of arbitrary order we demonstrate the efficiency and accuracy of the numerical method by providing a range of numerical simulations in particular we find that our approach can outperform the numerical methods considered in the literature by up to a factor of five although we focus on the kadomtsev petviashvili equation in this paper the proposed numerical scheme can be extended to a range of related models as well

families of varieties of general type over compact bases let be a smooth family of canonically polarized complex varieties over a smooth base generalizing the classical shafarevich hyperbolicity conjecture viehweg conjectured that y is necessarily of log general type if the family has maximal variation a somewhat stronger and more precise version of viehweg s conjecture was shown by the authors in s kebekus s j kovacs families of canonically polarized varieties over surfaces preprint math ag 0511378 invent math 2008 doi 10 1007 s00222 008 0128 8 s kebekus s j kovacs the structure of surfaces mapping to the moduli stack of canonically polarized varieties arxiv 0707 2054v1 math ag 2007 in the case where y is a quasi projective surface assuming that the minimal model program holds this very short paper proves the same result for projective base manifolds y of arbitrary dimension

well posedness for the fifth order shallow water equations abstract the cauchy problems for some kind of fifth order shallow water equations t u x 5 u x 3 u x u f u x u x 2 u 0 x t r r are considered by the fourier restriction norm method where nonlinear terms f u x u x 2 u are x u k k 2 3 u x 2 u or x u x 2 u respectively the local well posedness is established for data in h s r with s 7 4 for the kawahara equation f x u 2 and is established for data in h s r with s 1 4 for the modified kawahara equation f x u 3 respectively moreover the local result is established for data in h s r with s 0 if f u x 2 u and is established for data in h s r with s 1 4 if f x u x 2 u respectively

two exponential type integrators for the good boussinesq equation we introduce two exponential type integrators for the good boussinesq equation they are of orders one and two respectively and they require lower spatial regularity of the solution compared to classical exponential integrators for the first integrator we prove first order convergence in h r for solutions in h r 1 with r 1 2 this new integrator even converges with lower order in h r for solutions in h r i e without any additional smoothness assumptions for the second integrator we prove second order convergence in h r for solutions in h r 3 with r 1 2 and convergence in l 2 for solutions in h 3 numerical results are reported to illustrate the established error estimates the experiments clearly demonstrate that the new exponential type integrators are favorable over classical exponential integrators for initial data with low regularity

operator splitting for two dimensional incompressible fluid equations we analyze splitting algorithms for a class of two dimensional fluid equations which includes the incompressible navier stokes equations and the surface quasi geostrophic equation our main result is that the godunov and strang splitting methods converge with the expected rates provided the initial data are sufficiently regular

energy conserving hamiltonian boundary value methods for the numerical solution of the korteweg de vries equation abstract in this paper we study the efficient solution of the well known korteweg de vries equation equipped with periodic boundary conditions a fourier galerkin space semi discretization at first provides a large size hamiltonian ode problem whose solution in time is then carried out by means of energy conserving methods in the hbvm class hamiltonian boundary value methods the efficient implementation of the methods for the resulting problem is also considered and several numerical examples are reported

well posedness of the initial value problem for the korteweg de vries equation 1 1 itu axu u1xu o x t e r u x 0 uo x the kdv equation which was first derived as a model for unidirectional propagation of nonlinear dispersive long waves 21 has been considered in different contexts namely in its relation with the inverse scattering method in plasma physics and in algebraic geometry see 24 and references therein our purpose is to study local and global well posedness of the ivp 1 1 in classical sobolev spaces hs r we shall say that the ivp 1 1 is locally resp globally well posed in the function space x if it induces a dynamical system on x by generating a continuous local resp global flow it was established in the works of bona and smith 3 bona and scott 2 saut and temam 30 and kato 1 5 that the ivp 1 1 is locally resp globally well posed in hs with s 3 2 resp s 2 roughly speaking global well posedness in hs depends on the available local theory and on the conservation laws satisfied by solutions of 1 1 namely

operator splitting for the kdv equation we provide a new analytical approach to operator splitting for equations of the type u t au b u where a is a linear operator and b is quadratic a particular example is the korteweg de vries kdv equation u t uu x u xxx 0 we show that the godunov and strang splitting methods converge with the expected rates if the initial data are sufficiently regular

a splitting approach for the kadomtsev petviashvili equation we consider a splitting approach for the kadomtsev petviashvili equation with periodic boundary conditions and show that the necessary interpolation procedure can be efficiently implemented the error made by this numerical scheme is compared to exponential integrators which have been shown in klein and roidot 2011 2 to perform best for stiff solutions of the kadomtsev petviashvili equation since many classic high order splitting methods do not perform well we propose a stable extrapolation method in order to construct an efficient numerical scheme of order four in addition the conservation properties and the possibility of order reduction for certain initial values for the numerical schemes under consideration are investigated