dimension formula for graded lie algebras and its applications abstract in this paper we investigate the structure of infinite dimensional lie algebras l l l graded by a countable abelian semigroup satisfying a certain finiteness condition the euler poincare principle yields the denominator identities for the graded lie algebras from which we derive a dimension formula for the homogeneous subspaces l our dimension formula enables us to study the structure of the graded lie algebras in a unified way we will discuss some interesting applications of our dimension formula to the various classes of graded lie algebras such as free lie algebras kac moody algebras and generalized kac moody algebras we will also discuss the relation of graded lie algebras and the product identities for formal power series

perturbed collocation and runge kutta methods it is well known thatsome implicit runge kutta methods are equivalent to collocation methods this fact permits very short and natural proofs of order anda b an bn stability properties for this subclass of methods see 9 and 10 the present paper answers the natural question ifall rk methods can be considered as a somewhat perturbed collocation after having introduced this notion we give a proof on the order of the method and discuss their stability properties much of known theory becomes simple and beautiful

quantized enveloping algebras for borcherds superalgebras the rst author was supported in part by nsf grant dms 9300523 the second author was supported in part by the nondirected research fund korea research foundation 1996 the third author was supported in part by a faculty research grant from st lawrence university

exploring isospectral spring mass systems with firefly algorithm this paper investigates in line spring mass systems an fixed at one end and free at the other with n degrees of freedom d f the objective is to find feasible in line systems b n that are isospectral to a given system the spring mass systems a n and b n are represented by jacobi matrices an error function is developed with the help of the jacobi matrices a n and b n the problem of finding the isospectral systems is posed as an optimization problem with the aim of minimizing the error function the approach for creating isospectral systems uses the fact that the trace of two isospectral jacobi matrices a n and b n should be identical a modification is made to the diagonal elements of the given jacobi matrix a n to create the isospectral systems the optimization problem is solved using the firefly algorithm augmented by a local search procedure numerical results are obtained and resulting isospectral systems are shown for 4 d f and 10 d f systems

blockwise adaptivity for time dependent problems based on coarse scale adjoint solutions we describe and test an adaptive algorithm for evolution problems that employs a sequence of blocks consisting of fixed though nonuniform space meshes this approach offers the advantages of adaptive mesh refinement but with reduced overhead costs associated with load balancing remeshing matrix reassembly and the solution of adjoint problems used to estimate discretization error and the effects of mesh changes a major issue with a block adaptive approach is determining block discretizations from coarse scale solution information that achieve the desired accuracy we describe several strategies for achieving this goal using adjoint based a posteriori error estimates and we demonstrate the behavior of the proposed algorithms as well as several technical issues in a set of examples

effective solution of discontinuous ivps using a runge kutta formula pair with interpolants an automatic technique for solving discontinuous initial value problems is developed and justified the technique is based on the use of local interpolants such as those that have been developed for use with runge kutta formula pairs numerical examples are presented to illustrate the significant improvement in efficiency and reliability that results when this technique is used with standard methods

validated solutions of initial value problems for ordinary differential equations compared to standard numerical methods for initial value problems ivps for ordinary differential equations odes validated methods for ivps for odes have two important advantages if they return a solution to a problem then 1 the problem is guaranteed to have a unique solution and 2 an enclosure of the true solution is produced the authors survey taylor series methods for validated solutions of ivps for odes describe several such methods in a common framework and identify areas for future research

the monster lie algebra abstract we calculate the multiplicities of all the roots of the monster lie algebra this gives an example of a lie algebra all of whose simple roots and root multiplicities are known and which is not finite dimensional or an affine kac moody algebra there seem to be several similar infinite dimensional lie algebras which to a sympathetic eye appear to correspond to some of the sporadic simple groups

reliable approximate solution of systems of volterra integro differential equations with time dependent delays volterra integro differential equations with time dependent delay arguments can provide us with realistic models of many real world phenomena delayed lokta volterra predator prey systems arise in ecology and are well known examples of delay volterra integro differential equations dvides first introduced by volterra in 1928 we investigate the numerical solution of systems of dvides using an adaptive stepsize selection strategy we will present a generic variable stepsize approach for solving systems of neutral dvides based on an explicit continuous runge kutta method using defect error control and study the convergence of the resulting numerical method for various kinds of delay arguments we will show that the global error of the numerical solution can be effectively and reliably controlled by monitoring the size of the defect of the approximate solution and adjusting the stepsize on each step of the integration numerical results will be presented to demonstrate the effectiveness of this approach

global error estimation based on the tolerance proportionality for some adaptive runge kutta codes modern codes for the numerical solution of initial value problems ivps in odes are based in adaptive methods that for a user supplied tolerance d attempt to advance the integration selecting the size of each step so that some measure of the local error is email protected although this policy does not ensure that the global errors are under the prescribed tolerance after the early studies of stetter considerations concerning a theory for ode solvers in r burlisch r d grigorieff j schroder eds numerical treatment of differential equations proceedings of oberwolfach 1976 lecture notes in mathematics vol 631 springer berlin 1978 pp 188 200 tolerance proportionality in ode codes in r marz ed proceedings of the second conference on numerical treatment of ordinary differential equations humbold university berlin 1980 pp 109 123 and the extensions of higham global error versus tolerance for explicit runge kutta methods ima j numer anal 11 1991 457 480 the tolerance proportionality of adaptive ode solvers j comput appl math 45 1993 227 236 the reliability of standard local error control algorithms for initial value ordinary differential equations in proceedings the quality of numerical software assessment and enhancement ifip series springer berlin 1997 it has been proved that in many existing explicit runge kutta codes the global errors behave asymptotically as some rational power of d this step size policy for a given ivp determines at each grid point t n a new step size h n 1 h t n d so that h t d is a continuous function of t in this paper a study of the tolerance proportionality property under a discontinuous step size policy that does not allow to change the size of the step if the step size ratio between two consecutive steps is close to unity is carried out this theory is applied to obtain global error estimations in a few problems that have been solved with the code gauss2 s gonzalez pinto r rojas bello gauss2 a fortran 90 code for second order initial value problems based on an adaptive two stage runge kutta gauss method with this discontinuous step size policy

the tolerance proportionality of adaptive ode solvers abstract modern software for solving ordinary differential equation ode initial value problems requires the user to specify the ode and choose a value for the error tolerance the software can be thought of as a black box with a dial turning the dial changes the accuracy and expense of the integration process it is therefore of interest to know how the global error varies with the error tolerance in this work we look at explicit runge kutta methods and show that with any standard error control method and ignoring higher order terms the global error in the numerical solution behaves like a known rational power of the error tolerance this generalises earlier work of stetter who found sufficient conditions for the global error to be linear in the tolerance we also display the order of the next highest term we then analyse continuous runge kutta schemes and show what order of interpolation is necessary and sufficient for the continuous approximation to inherit the tolerance proportionality of the discrete formula finally we extend the results to the case of ode systems with constant delays thereby generalising some previous results of the author

monstrous moonshine and monstrous lie superalgebras we prove conway and norton s moonshine conjectures for the infinite dimensional representation of the monster simple group constructed by frenkel lepowsky and meurman to do this we use the no ghost theorem from string theory to construct a family of generalized kac moody superalgebras of rank 2 which are closely related to the monster and several of the other sporadic simple groups the denominator formulas of these superalgebras imply relations between the thompson functions of elements of the monster i e the traces of elements of the monster on frenkel lepowsky and meurman s representation which are the replication formulas conjectured by conway and norton these replication formulas are strong enough to verify that the thompson functions have most of the moonshine properties conjectured by conway and norton and in particular they are modular functions of genus 0 we also construct a second family of kac moody superalgebras related to elements of conway s sporadic simple group col these superalgebras have even rank between 2 and 26 for example two of the lie algebras we get have ranks 26 and 18 and one of the superalgebras has rank 10 the denominator formulas of these algebras give some new infinite product identities in the same way that the denominator formulas of the a n n e kac moody algebras give the macdonald identities

a generalization of borcherds algebra and denominator formula abstract we study generalized lie superalgebras an extension of kac s generalized lie superalgebras and show that they are transformed forms ofl graded lie superalgebras for some abelian groupl we then introduce a generalized lie superalgebra version of the generalized kac moody algebra borcherds algebra since it is a transformed borcherds superalgebra it has several properties similar to those of borcherds superalgebra for example it is defined by similar relations and has a similar denominator formula and character formulas

siegel automorphic form corrections of some lorentzian kac moody lie algebras we find automorphic form corrections which are generalized lorentzian kac moody superalgebras without odd real simple roots for two elliptic lorentzian kac moody algebras of rank 3 with a lattice weyl vector and calculate multiplicities of their simple and arbitrary roots these kac moody algebras are defined by hyperbolic symmetrized generalized cartan matrices inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i of rank 3 both these algebras have elliptic type i e their weyl groups have fundamental polyhedra of finite volume in corresponding hyperbolic spaces and have a lattice weyl vector the correcting automorphic forms are siegel modular forms the form corresponding to g 1 is the classical siegel cusp form of weight 5 which is the product of ten even theta constants in particular we find an infinite product formula for this modular form

adaptive stiff solvers at low accuracy and complexity this paper is concerned with adaptive stiff solvers at low accuracy and complexity for systems of ordinary differential equations the considered stiff solvers are two second order rosenbrock methods with low complexity and the bdf method of the same order for the adaptive algorithm we propose to use a monitor function defined by comparing a measure of the local variability of the solution times the used step size and the order of magnitude of the solution instead of the classical approach based on some local error estimation this simple step size selection procedure is implemented in order to control the behavior of the numerical solution it is easily used to automatically adjust the step size as the calculation progresses until user specified tolerance bounds for the introduced monitor function are fulfilled this leads to important advantages in accuracy efficiency and general easier of use at the end of the paper we present two numerical tests which show the performance of the implementation of the stiff solvers with the proposed adaptive procedure

what good are numerical simulations of chaotic dynamical systems abstract numerical simulations of mathematical models can suggest that the models are chaotic for example one can compute an orbit and its associated finite time lyapunov exponents and these computed exponents can be positive it is not clear how far these suggestions can be trusted because as is well known numerical methods can introduce spurious chaos or even suppress actual chaos this focused review examines the fidelity of numerical methods we look at the didactic example of the gauss map from the theory of continued fractions which allows a simple examination of backward error analysis for discrete dynamical systems and gives a clear picture of the effects of floating point arithmetic a similar use of backward error analysis in the form of defect control gives a useful understanding in the case of continuous dynamical systems finally we discuss limitations of this backward point of view

the fake monster superalgebra abstract we show that the physical states of a 10 dimensional superstring moving on a torus form a generalized kac moody superalgebra this gives the first explicit realizations of these algebras for a special torus the denominator function of this algebra is an automorphic form so that we can determine the simple roots we call this algebra the fake monster superalgebra

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

defect controlled numerical methods and shadowing for chaotic differential equations abstract it is well known that chaotic ode s are in the classical sense unstable or ill conditioned it is not clear that variable step size codes based on local error control can solve such problems in a useful way in particular it is usually difficult to show that the apparent chaos in the resulting solution is not a numerical artifact i show here that a defect controlled method gives useful solutions for chaotic problems a pragmatically modified definition of what it means for a dynamical system to be chaotic is also presented