interpolation for runge kutta methods runge kutta methods provide a popular way to solve the initial value problem for a system of ordinary differential equations in contrast to the adams methods there is no natural way to approximate the solution between mesh points a way to accomplish this is proposed which is applicable to some important formulas its theoretical support is much better than that of interpolation in the popular variable order variable step adams codes

numerical solution of delay differential equations by uniform corrections to an implicit runge kutta method in this paper we develop a class of numerical methods to approximate the solutions of delay differential equations they are essentially based on a modified version in a predictor corrector mode of the one step collocation method atn gaussian points these methods applied to ode s provide a continuous approximate solution which is accurate of order 2n at the nodes and of ordern 1 uniformly in the whole interval in order to extend the methods to delay differential equations the uniform accuracy is raised to the order 2n by some a posteriori corrections numerical tests and comparisons with other methods are made on real life problems

the frobenius semiradical of a lie algebra univ limburg dept math b 3590 diepenbeek belgium ooms ai univ limburg dept math univ campus b 3590 diepenbeek belgium alfons ooms luc ac be

on the asymptotic stability properties of runge kutta methods for delay differential equations in this paper asymptotic stability properties of runge kutta r k methods for delay differential equations ddes are considered with respect to the following test equation begin eqnarray left begin array ccl y t a y t b y t tau 0 3cm y t g t end array right hskip 0 3cm begin array r t 0 0 3cm tau leq t leq 0 end array end eqnarray where a b in bbb r tau 0 and g t is a continuous real valued function in the last few years stability properties of r k methods applied to ddes have been studied by numerous authors who have considered regions of asymptotic stability for any positive delay and thus independent of the specific value of tau

a sufficient condition for gpn stability for delay differential equations in this paper the author considers a linear test delay differential equation with non constant coefficients related to the definitions of pn and gpn stability for numerical methods he defines stability properties for an ordinary differential equation together with stability properties of interpolants for numerical methods and in this way he gives sufficient conditions for gpn stability

comparison of the asymptotic stability for multirate rosenbrock methods systems of ordinary differential equations containing different time scales can be treated by multirate extensions of singlerate integration methods since the stability properties of a singlerate method are usually not carried over to the corresponding multirate method it is necessary to investigate the asymptotic stability of multirate methods we apply different linearly implicit rosenbrock methods in a recursive multirate procedure for the interpolation continuous extension hermite interpolation and a special monotone hermite interpolation are considered we give stability regions and illustrate their importance for a two component test problem

construction of a multirate rodas method for stiff odes multirate time stepping is a numerical technique for efficiently solving large scale ordinary differential equations odes with widely different time scales localized over the components this technique enables one to use large time steps for slowly varying components and small steps for rapidly varying ones multirate methods found in the literature are normally of low order one or two focusing on stiff odes in this paper we discuss the construction of a multirate method based on the fourth order rodas method special attention is paid to the treatment of the refinement interfaces with regard to the choice of the interpolant and the occurrence of order reduction for stiff linear systems containing a stiff source term we propose modifications for the treatment of the source term which overcome order reduction originating from such terms and which we can implement in our multirate method

primitive group rings and noetherian rings of quotients let k be a field and let g be a countable nilpotent group with centre z we show that the group algebra kg is primitive if and only if k is countable g is torsion free and there exists an abelian subgroup a of g of infinite rank with a fn z 1 suppose now that g is torsion free then kg has a partial quotient ring q kg kz 1 the above characterisation of the primitivity of kg is intimately connected with the question when is q a noetherian ring we determine this for those groups g as above all of whose finite rank subgroups are finitely generated in this case q is noetherian if and only if g has no abelian subgroup a of infinite rank with a fn z 1

on the center and semi center of enveloping algebras in prime characteristic abstract theorems of j dixmier and c moeglin on the semi center of enveloping algebras in zero characteristic and of r p stanley and h nakajima on polynomial relative invariants of finite groups are shown to have analogs for enveloping algebras and for lie algebra polynomial invariants in the prime characteristic case we shall illustrate by examples the extent of these analogies a key result is the realization of the semi center as a fixed ring of the action of a finite set of nilpotent derivations on the center of an enveloping algebra of a related lie algebra

one step collocation uniform superconvergence predictor corrector method local error estimate in this paper we consider the one step collocation method at gaussian points for the numerical solution of ivp s for ode s it is well known that this method has a superconvergence rate o h 2n at the nodes while it uniformly converges with the lower order o h n 1 we give a rule to reach the uniform superconvergence by some kind of idec method an interesting by product of this procedure is an accurate estimation of the local discretization error moreover we give a modified version of the method in a predictor corrector mode for which all the previous results keep on holding in particular we propose a new method of order four with step size control

natural continuous extensions of runge kutta methods the present paper develops a theory of natural continuous extensions nces for the discrete approximate solution of an ode given by a runge kutta process these nces are defined in such a way that the continuous solutions furnished by the one step collocation methods are included

continuous parallel block methods and their applications continuous numerical methods have many applications in the numerical solution of discontinuous ordinary differential equations delay differential equations delay differential algebraic equations neutral delay differential equations integro differential equations etc this paper deals with a continuous extension for the discrete approximate solution of ordinary differential equations generated by a class of parallel block methods existence and uniqueness for the continuous extension are discussed convergence and absolute stability of the continuous parallel block methods for ordinary differential equations are studied as applications we adopt the continuous parallel block methods to solve delay differential equations neutral delay differential equations and delay differential algebraic equations and obtain sufficient and necessary conditions for the continuous parallel block methods to be asymptotically stable several numerical experiments are given to illustrate the performance of the continuous parallel block methods

order barriers for continuous explicit runge kutta methods in this paper we deal with continuous numerical methods for solving initial value problems for ordinary differential equations the need for which occurs frequently in applications whereas most of the commonly used multistep methods provide continuous extensions by means of an interpolant which is available without making extra function evaluations this is not always the case for one step methods we consider the class of explicit runge kutta methods and provide theorems used to obtain lower bounds for the number of stages required to construct methods of a given unifonn order p these bounds are similar to the butcher barriers known for the discrete case and are derived up to order p 5 as far as we know the examples we present of 8 stage continuous runge kutta methods of uniform order 5 are the first of their kind

natural continuous extensions of runge kutta methods for volterra integral equations of the second kind and their applications we consider a very general class of runge kutta methods for the numerical solution of volterra integral equations of the second kind which includes as special cases all the more important methods which have been considered in the literature the main purpose of this paper is to define and prove the existence of the natural continuous extensions nce s of runge kutta methods i e piecewise polynomial functions which extend the approximation at the grid points to the whole interval of integration the particular properties required of the nce s allow us to construct the tail approximations which are quite efficient in terms of kernel evaluations

multistep high order interpolants of runge kutta methods abstract we consider a p order runge kutta method k n i x n c i h y n h j 1 a ij k n j i 1 y n 1 y n h i 1 b i k n i for solving an initial value problem for ordinary differential equations the aim of this paper is to construct p order interpolants by using the values furnished by the method on n successive intervals of integration by using lagrange interpolation one can obtain a p order interpolant over p intervals but we are interested in finding the minimum number of intervals needed to obtain this we provide the conditions to be satisfied and we obtain an estimation of the number n some examples are given

the polynomiality of the poisson center and semi center of a lie algebra and dixmier s fourth problem abstract let g be a finite dimensional lie algebra over an algebraically closed field k of characteristic zero we provide necessary and also some sufficient conditions in order for its poisson center and semi center to be polynomial algebras over k this occurs for instance if g is quadratic of index 2 with g g g and also if g is nilpotent of index at most 2 the converse holds for filiform lie algebras of type l n q n r n and w n we show how dixmier s fourth problem for an algebraic lie algebra g can be reduced to that of its canonical truncation g moreover dixmier s statement holds for all lie algebras of dimension at most eight the nonsolvable ones among them possess a polynomial poisson center and semi center

on commutative polarizations let l be a finite dimensional lie algebra over a field k of characteristic zero and let u l be its enveloping algebra with quotient division ring d l let p be a commutative lie subalgebra of l in o2 the necessary and sufficient condition on p was given in order for d p to be a maximal commutative subfield of d l in particular this condition is satisfied if p is a commutative polarization cp with respect to any regular f l and the converse holds if l is ad algebraic the purpose of this paper is to study lie algebras admitting these cp s and to demonstrate their widespread occurrence first we have the following characterisation if l is completely solvable p is a cp of l if and only if there exists a descending chain of lie subalgebras

efficient solution of nonlinear ordinary differential equations expressed in s system canonical form ordinary differential equations can be recast into a nonlinear canonical form called an s system evidence for the generality of this class comes from extensive empirical examples that have been recast and from the discovery that sets of differential equations and functions recognized as among the most general are special cases of s systems identification of this nonlinear canonical form suggests a radically different approach to numerical solution of ordinary differential equations by capitalizing on the regular structure of s systems efficient formulas for a variable order variable step taylor series method are developed the minimum improvement in efficiency over conventional runge kutta methods is shown to be more than 20 to 60 percent for equations expressed in s system form tests with implemented methods demonstrate that 10 to 100 fold reductions in time are actually realized for solution of s systems to examine the potential benefit of recasting ordinary differential equations and then sol

the gelfand kirillov conjecture for semi direct products of lie algebras let g be an n dimensional lie algebra over a field k of characteristic zero and let w be a g module such that dimw n sufficient conditions are given in order for the semi direct product g w to satisfy the gelfand kirillov conjecture this implies that this conjecture holds for an important class of frobenius lie algebras special attention is devoted to the case where g sl 2 k

fourth and fifth order scaled rungs kutta algorithms for treating dense output new runge kutta algorithms are developed which determine the solution of a system of ordinary differential equations at any point within a given integration step as well as at the end of each step

natural continuous extensions of runge kutta methods for volterra integrodifferential equations in this paper we deal with a very general class of runge kutta methods for the numerical solution of volterra integrodifferential equations our main contribution is the development of the theory of natural continuous extensions nces i e piecewice polynomial functions which interpolate the values given by the rk method at the mesh points the particular features of these nces allow us to construct tail approximations which are quite efficient since they require a minimal number of kernel evaluations

the gelfand kirillov conjecture for lie algebras of dimension at most eight univ reims dept math f 51687 reims 2 france univ limburg dept math b 3590 diepenbeek belgium alev j univ reims dept math bp 1039 f 51687 reims 2 france

computing invariants and semi invariants by means of frobenius lie algebras abstract let u be the enveloping algebra of a finite dimensional lie algebra over a field k of characteristic zero z u its center and sz u its semi center a sufficient condition is given in order for sz u to be a polynomial algebra over k surprisingly this condition holds for many lie algebras especially among those for which the radical is nilpotent in which case sz u z u in particular it allows the explicit description of z u for more than half of all complex indecomposable nilpotent lie algebras of dimension at most 7

on certain maximal subfields in the quotient division ring of an enveloping algebra univ limburg dept math b 3590 diepenbeek belgium ooms ai univ limburg dept math b 3590 diepenbeek belgium

the poisson center and polynomial maximal poisson commutative subalgebras especially for nilpotent lie algebras of dimension at most seven abstract let g be a finite dimensional lie algebra over an algebraically closed field k of characteristic zero we collect some general results on the poisson center of s g including some simple criteria regarding its polynomiality and also on certain poisson commutative subalgebras of s g these facts are then used to complete our previous work on the subject ooms 2009 o5 5 i e to give an explicit description for the poisson center of all indecomposable nilpotent lie algebras of dimension at most seven among other things we also provide a polynomial maximal poisson commutative subalgebra of s g enjoying additional properties as a by product we show that a conjecture by milovanov is valid in this situation these results easily carry over to the enveloping algebra u g

symmetric invariant non degenerate bilinear form on a lie algebra the killing form is a powerful tool in the study of semi simple lie algebras unfortunately it is non degenerate only in the semi simple case therefore one has to look more generally for a form which has all the useful properties of the killing form bilinearity symmetry invariance nondegeneracy such a form will be a scalar multiple of the killing form in the central simple case but in the general case it may not even exist just take the solvable lie algebra of dimension 2 in this work we study what we call regular quadratic lie algebra g cp i e lie algebra g equipped with a symmetric invariant non degenerate bilinear form cp if d is a derivation of g belonging to the orthogonal lie algebra o cp then following the ideas of v g kac we define a regular quadratic lie algebra of dimension 2 dim g denoted g pi which we call the extension of g cp by d proposition 2 3 the interesting point of this extension is the fact that any indecomposable regular quadratic lie algebra with nontrivial center is of the form g pi proposition 2 9 which allows an inductive classification moreover a regular quadratic solvable lie algebra has a non zero center lemma 2 8 therefore the solvable case is completely covered proposition 2 11 classified all extensions of a given g cp

on the semi center of a universal enveloping algebra throughout this paper l will be a nonzero finite dimensional lie algebra over a field k of characteristic zero let u l be the universal enveloping algebra of l with center z u l d l the division ring of quotients of u l with center z d l for each 2 e l we denote by d l the set of those ued l such that ad x u n x u for all xe l its elements are called the semi invariants of d l relative to 1 clearly d l l c al a for all i p e l the sum of the d l is direct and is a subalgebra sz d l of d l which is called the semi center of d l we put u l u l n d l the sum sz u l of the u l is called the semi center of u l the semi center sz s l of the symmetric algebra s l of l can be defined in a similar fashion the semi center is a commutative subalgebra of u l which never reduces to k 7 and which is stable under all automorphisms of u l 16 from the fact that it is a unique factorization domain in the algebraically closed case 131 we are able to deduce that in the general case sz u l is contained in a factorial subalgebra of u l theorem 1 2 in particular any nonzero semi invariant of u l can be factored in a unique way as the product of irreducible semiinvariants proposition 1 5 this result proves to be extremely useful throughout the paper we also obtain some characterizations of the semiinvariants of d l proposition 1 8 let a l resp a l be the set of all z e l such that u l 0 resp d l o a l is a semi group which is not always finitely generated 9 the group n l which is isomorphic to kharchenko s group of x inner automorphisms of u l 16 turns out to be the additive subgroup of l generated by a l corollary 1 9 next the centralizer of sz u l in u l coincides with u l where l is the intersection of ker 2 ac l corollary 1 16 l 324 0021 8693 85 3 00

a degree inequality for lie algebras with a regular poisson semi center for lie algebras whose poisson semi center is a polynomial ring we give a bound for the sum of the degrees of the generating semi invariants this bound was previously known in many special cases

stability properties of interpolants for runge kutta methods in solving initial value problems for ordinary differential equations by means of a discrete method we often need to know the solution on a set of points that differs from the grid systems with driving equation dense output discontinuity problems delay equations etc the nonnodal approximations are generally obtained by local interpolation and different features can be requested from the interpolants according to the problem treated in this paper we introduce the concept of stable interpolant for runge kutta methods roughly speaking a stable interpolant maintains the stability properties of the runge kutta method itself the lack of such stability makes the continuous extension not reliable especially in solving stiff equations