the numerical analysis of ordinary differential equations runge kutta and general linear methods mathematical and computational introduction the euler method and its generalizations analysis of runge kutta methods general linear methods bibliography

structure and stability of online chat networks built on emotion carrying links abstract high resolution data of online chats are studied as a physical system in the laboratory in order to quantify collective behavior of users our analysis reveals strong regularities characteristic of natural systems with additional features in particular we find self organized dynamics with long range correlations in user actions and persistent associations among users that have the properties of a social network furthermore the evolution of the graph and its architecture with specific k core structure are shown to be related with the type and the emotion arousal of exchanged messages partitioning of the graph by deletion of the links which carry high arousal messages exhibits critical fluctuations at the percolation threshold

an algebraic approach to invariant preserving integators the case of quadratic and hamiltonian invariants in this article conditions for the preservation of quadratic and hamiltonian invariants by numerical methods which can be written as b series are derived in a purely algebraical way the existence of a modified invariant is also investigated and turns out to be equivalent up to a conjugation to the preservation of the exact invariant a striking corollary is that a symplectic method is formally conjugate to a method that preserves the hamitonian exactly another surprising consequence is that the underlying one step method of a symmetric multistep scheme is formally conjugate to a symplectic p series when applied to newton s equations of motion

hierarchical sequencing of online social graphs abstract in online communications patterns of conduct of individual actors and use of emotions in the process can lead to a complex social graph exhibiting multilayered structure and mesoscopic communities using simplicial complexes representation of graphs we investigate in depth topology of the online social network constructed from myspace dialogs which exhibits original community structure a simulation of emotion spreading in this network leads to the identification of two emotion propagating layers three topological measures are introduced referred to as the structure vectors which quantify graph s architecture at different dimension levels notably structures emerging through shared links triangles and tetrahedral faces frequently occur and range from tree like to maximal 5 cliques and their respective complexes on the other hand the structures which spread only negative or only positive emotion messages appear to have much simpler topology consisting of links and triangles the node s structure vector represents the number of simplices at each topology level in which the node resides and the total number of such simplices determines what we define as the node s topological dimension the presented results suggest that the node s topological dimension provides a suitable measure of the social capital which measures the actor s ability to act as a broker in compact communities the so called simmelian brokerage we also generalize the results to a wider class of computer generated networks investigating components of the node s vector over network layers reveals that same nodes develop different socio emotional relations and that the influential nodes build social capital by combining their connections in different layers

backward error analysis for multistep methods in recent years much insight into the numerical solution of ordinary differential equations by one step methods has been obtained with a backward error analysis it allows one to explain interesting phenomena such as the almost conservation of energy the linear error growth in hamiltonian systems and the existence of periodic solutions and invariant tori in the present article the formal backward error analysis as well as rigorous exponentially small error estimates are extended to multistep methods a further extension to partitioned multistep methods is outlined and numerical illustrations of the theoretical results are presented

the control of parasitism in g symplectic methods g symplectic general linear methods are designed to approximately preserve symplectic invariants for hamiltonian systems in this paper the properties of g symplectic methods are explored computationally and theoretically good preservation properties are observed over long times for many parameter ranges but for other parameter values the parasitic behavior to which multivalue methods are prone corrupts the numerical solution by the growth of small perturbations two approaches for alleviating this effect are considered first compositions of methods with growth parameters of opposite signs can be used to cancel the long term effect of parasitism second methods can be constructed for which the growth parameters are zero by design each of these remedies is found to be successful in eliminating parasitic behavior in long term simulations using a variety of test problems

lie algebras graded by finite root systems and intersection matrix algebras this paper classifies the lie algebras graded by doubly laced finite root systems and applies this classification to identify the intersection matrix algebras arising from multiply affinized cartan matrices of types b c f and g this completes the determination of the lie algebras graded by finite root systems initiated by berman and moody who studied the simply laced finite root systems of rank 2

the averaged lagrangian method in this paper we present a new class of symplectic methods namely variational integrators employing the same integral averaging found in the averaged vector field method extensions to higher order are obtained using a collocation approach the obtained methods are time symmetric p series methods the proposed methods are compared with existing integrators the tests focus on highly oscillatory problems like the fermi pasta ulam problem

new energy preserving algorithms for nonlinear hamiltonian wave equation equipped with neumann boundary conditions abstract in this paper using the blend of spatial discretization by second order or fourth order finite difference methods fdm and time integration by the generalized average vector field gavf method or the generalized adapted average vector field gaavf method we propose and analyze novel energy preserving algorithms for solving the nonlinear hamiltonian wave equation equipped with homogeneous neumann boundary conditions firstly two kinds of finite difference methods are considered to discretize the spatial derivative which can be of order two and order four respectively in all the spatial grid points the conservation laws of the discrete energy are established after the semi discretization a hamiltonian system of odes is derived whose hamiltonian can be regarded as the approximate energy of the original continuous system then the gavf formula and the gaavf formula are developed and applied to the derived hamiltonian odes to yield some novel and efficient algorithms which can exactly preserve the discrete energy the numerical simulation is implemented and the numerical results demonstrate the spatial and temporal accuracy and the remarkable energy preserving property of the new algorithms presented in this paper

transmission of packets on a hierarchical network statistics and explosive percolation we analyze an idealized model for the transmission or flow of particles or discrete packets of information in a weight bearing branching hierarchical two dimensional network and its variants the capacities add hierarchically down the clusters each node can accommodate a limited number of packets depending on its capacity and the packets hop from node to node following the links between the nodes the statistical properties of this system are given by the maxwell boltzmann distribution we obtain analytical expressions for the mean occupation numbers as functions of capacity for different network topologies the analytical results are shown to be in agreement with the numerical simulations the traffic flow in these models can be represented by the site percolation problem it is seen that the percolation transitions in the 2d model and in its variant lattices are continuous transitions whereas the transition is found to be explosive discontinuous for the v lattice the critical case of the 2d lattice the scaling behavior of the second order percolation case is studied in detail we discuss the implications of our analysis

percolation in directed scale free networks many complex networks in nature have directed links a property that affects the network s navigability and large scale topology here we study the percolation properties of such directed scale free networks with correlated in and out degree distributions we derive a phase diagram that indicates the existence of three regimes determined by the values of the degree exponents in the first regime we regain the known directed percolation mean field exponents in contrast the second and third regimes are characterized by anomalous exponents which we calculate analytically in the third regime the network is resilient to random dilution i e the percolation threshold is pc 1

canonical b series b series provide a powerful general tool to express numerical methods for differential equations many differential equations are of hamiltonian form and there has been much recent interest in constructing so called canonical or symplectic integrators for the hamiltonian case in this paper we provide a necessary and sufficient condition for a b series to correspond to a canonical method

multi step methods are essentially one step methods in this note the geometry of multi step methods is studied using invariant manifold theory for maps as familiar from dynamical systems theory this permits to associate a one step method to each multi step method to which the former is not only equivalent asymptotically but equal in each step if the one step method is used to produce the initial data of the multi step method

energy and quadratic invariants preserving integrators based upon gauss collocation formulae we introduce a new family of symplectic integrators for canonical hamiltonian systems each method in the family depends on a real parameter alpha when alpha 0 we obtain the classical gauss collocation formula of order 2s where s denotes the number of the internal stages for any given non null alpha the corresponding method remains symplectic and has order 2s 2 hence it may be interpreted as an o h 2s 2 symplectic perturbation of the gauss method under suitable assumptions we show that the parameter alpha may be properly tuned at each step of the integration procedure so as to guarantee energy conservation in the numerical solution as well as to maintain the original order 2s as the generating gauss formula

the lack of continuity and the role of infinite and infinitesimal in numerical methods for odes the case of symplecticity when numerically integrating canonical hamiltonian systems the long term conservation of some of its invariants for example the hamiltonian function itself assumes a central role the classical approach to this problem has led to the definition of symplectic methods among which we mention gauss legendre collocation formulae indeed in the continuous setting energy conservation is derived from symplecticity via an infinite number of infinitesimal contact transformations however this infinite process cannot be directly transferred to the discrete setting by following a different approach in this paper we describe a sequence of methods sharing the same essential spectrum and then the same essential properties which are energy preserving starting from a certain element of the sequence on i e after a finite number of steps

quadratic invariants and multi symplecticity of partitioned runge kutta methods for hamiltonian pdes in this paper we study the preservation of quadratic conservation laws of runge kutta methods and partitioned runge kutta methods for hamiltonian pdes and establish the relation between multi symplecticity of runge kutta method and its quadratic conservation laws for schrodinger equations and dirac equations it reveals that multi symplectic runge kutta methods applied to equations with appropriate boundary conditions can preserve the global norm conservation and the global charge conservation respectively

b series methods are exactly the affine equivariant methods butcher series also called b series are a type of expansion fundamental in the analysis of numerical integration numerical methods that can be expanded in b series are defined in all dimensions so they correspond to sequences of maps one map for each dimension a long standing problem has been to characterise those sequences of maps that arise from b series this problem is solved here we prove that a sequence of smooth maps between vector fields on affine spaces has a b series expansion if and only if it is affine equivariant meaning it respects all affine maps between affine spaces

hopf algebras renormalization and noncommutative geometry we explore the relation between the hopf algebra associated to the renormalization of qft and the hopf algebra associated to the ncg computations of tranverse index theory for foliations

geometric integrators for odes geometric integration is the numerical integration of a differential equation while preserving one or more of its geometric properties exactly i e to within round off error many of these geometric properties are of crucial importance in physical applications preservation of energy momentum angular momentum phase space volume symmetries time reversal symmetry symplectic structure and dissipation are examples in this paper we present a survey of geometric numerical integration methods for ordinary differential equations our aim has been to make the review of use for both the novice and the more experienced practitioner interested in the new developments and directions of the past decade to this end the reader who is interested in reading up on detailed technicalities will be provided with numerous signposts to the relevant literature

generalized kac moody algebras on etudie une classe d algebres de lie qui ont une forme bilineaire contravariante qui est presque definie positive ces algebres generalisent celles de kac moody et peuvent etre considerees comme des algebres de kac moody avec des racines simples imaginaires

lie algebras graded by finite root systems and the intersection matrix algebras of slodowy this paper is about toroidal lie algebras certain intersection matrix lie algebras defined by slodowy and their relationship to one another and to certain lie algebra analogues of steinberg groups the main result of the paper is the identification of the intersection matrix algebras arising from multiply affinized cartan matrices of types a d and e with certain steinberg lie algebras and toroidal lie algebras propositions 5 9 and 5 10 a major part of the paper studies and classifies lie algebras graded by finite root systems these become the princi pal tool in our analysis of intersection matrix algebras each lie algebra graded by a simply laced finite root system of rank 2 has attached to it an algebra which according to the type and rank is either commutative and associative only associative or alternative all these possibilities occur in our description of inter section matrix algebras let r be any associative algebra with identity not necessarily finite dimen sional over a field k of characteristic 0 for each positive integer n the associative algebra m r of n n matrices with coefficients in r forms a lie algebra over k under the commutator product we denote this lie algebra by ol r let eis be the i j matrix unit of m r and assume that n 2 the subalgebra e r of oi r generated by the elements reis r r i 4 j is an ideal of 91 r and is perfect i e it is its own derived algebra now any perfect lie algebra o has a universal central extension also perfect called a universal covering algebra u c a of o ga so in particular e r has a u c a that we will denote by t r we define 112 r by the exact sequence 0 1 0 f2 r t r e r 0 dedicated to our teacher maria j wonenburger both authors gratefully acknowledge the support of the natural sciences and engineering research council of canada

lie algebras graded by 3 graded root systems and jordan pairs covered by grids let r be an irreducible root system a lie algebra l is called graded by r if l is graded with grading group the root lattice of r such that the only nonzero homogeneous subspaces of l have degree 0 or a root in r the grading is induced by the adjoint action of a split cartan subalgebra of a finite dimensional simple lie subalgebra of l with root system r and l is generated by the homogeneous subspaces of nonzero degree this class of lie algebras was introduced and studied by s berman and r moody in invent math 108 1992 where in particular a classification up to central equivalence is given in the simply laced case the doubly laced cases have recently been classified by g benkart and e zelmanov let r be a 3 graded root system i e r is not of type e 8 f 4 or g 2 in this paper lie algebras graded by r are described in a unified way without case by case considerations namely it is shown that a lie algebra l is 3 graded if and only if l is a central extension of the tits kantor koecher algebra of a jordan pair covered by a grid whose associated 3 graded root system is isomorphic to r this result is then used to classify lie algebras graded by r we give the classification of jordan pairs covered by a grid and describe their tits kantor koecher algebras one of the advantages of this approach is that it works over rings containing 1 2 and 1 3 and also for infinite root systems another application is the description of slodowy s intersection matrix algebras arising from multiply affinized cartan matrices

symmetric multistep methods over long times for computations of planetary motions with special linear multistep methods an excellent long time behaviour is reported in the literature without a theoretical explanation neither the total energy nor the angular momentum exhibit secular error terms in this paper we completely explain this behaviour by studying the modified equation of these methods and by analyzing the remarkably stable propagation of parasitic solution components

generalized percolation in random directed networks we develop a general theory for percolation in directed random networks with arbitrary two point correlations and bidirectional edges that is edges pointing in both directions simultaneously these two ingredients alter the previously known scenario and open new views and perspectives on percolation phenomena equations for the percolation threshold and the sizes of the giant components are derived in the most general case we also present simulation results for a particular example of uncorrelated network with bidirectional edges confirming the theoretical predictions

a galerkin energy preserving method for two dimensional nonlinear schrodinger equation abstract in this paper a galerkin energy preserving scheme is proposed for solving nonlinear schrodinger equation in two dimensions the nonlinear schrodinger equation is first rewritten as an infinite dimensional hamiltonian system following the method of lines the spatial derivatives of the nonlinear schrodinger equation are approximated with the aid of the galerkin methods the resulting ordinary differential equations can be cast into a canonical hamiltonian system a fully discretized scheme is then devised by considering an average vector field method in time moreover based on the fast fourier transform and the matrix diagonalization method a fast solver is developed to solving the resulting algebraic equations finally the proposed scheme is employed to capture the blow up phenomena of the nonlinear schrodinger equation