accelerated monotone scheme for finite difference equations concerning steady state prey predator interactions abstract in previous work by adapting a suitable finite difference method to a particular monotone scheme the authors and a lazer have studied the numerical solution of a system of semilinear elliptic partial differential equations which determines the equilibria of the volterra lotka equations describing prey predator interactions with diffusion in this paper in order to improve the efficiency of the method we show how newton s method can be successfully combined with the previous scheme to greatly accelerate the convergence in some particularly difficult problems the new method reduces the average number of iterations necessary to generate each element of the monotone sequences from 15 to about 3

the decomposition of lie powers let g be a group f a field of prime characteristic p and v a finite dimensional fg module let l v denote the free lie algebra on v regarded as an fg submodule of the free associative algebra or tensor algebra t v for each positive integer r let l r v and t r v be the r th homogeneous components of l v and t v respectively here l r v is called the r th lie power of v our main result is that there are submodules b 1 b 2 of l v such that for all r b r is a direct summand of t r v and whenever m geqslant 0 and k is not divisible by p the module l p mk v is the direct sum of l p m b k l p m 1 b pk l 1 b p mk thus every lie power is a direct sum of lie powers of p power degree the approach builds on an analysis of t r v as a bimodule for g and the solomon descent algebra

monotone scheme for finite difference equations concerning steady state prey predator interactions abstract in this article a system of semilinear elliptic partial differential equations is studied this system determines the equilibria of the volterra lotka equations describing prey predator interactions with diffusion to analyze the system a new monotone scheme is presented a rigorous foundation is given for numerical calculations by adapting a suitable finite difference method to the new monotone scheme earlier theories in finite differences are not successful in solving the system without this scheme

smart 2017 second conference on subdivision geometric and algebraic methods isogeometric analysis and refinability in italy abstract this introductory paper provides an overview of a selection of contributions presented at the 2nd international conference on subdivision geometric and algebraic methods isogeometric analysis and refinability in italy smart2017 gaeta september 17 21 2017 the smart2017 special issue is also an opportunity to honor the memory of laura gori who was one of the proposers of this conference series

accelerated monotone iterations for numerical solutions of nonlinear elliptic boundary value problems abstract an accelerated monotone iterative scheme for numerical solutions of a class of nonlinear elliptic boundary value problems is presented the mathematical analysis is devoted to a system of discretized equations of the elliptic boundary value problem by the finite difference or finite element method it is shown that the sequence of iterations from a linear iteration process converges monotonically and quadratically to a unique solution in a sector between a pair of upper and lower solutions this result is then used to show the quadratic convergence of the iterations to a maximal solution and a minimal solution when the nonlinear discrete system possesses multiple solutions an application is given to a tabular reactor model from chemical engineering for numerical solutions and the number of iterations are compared with that by the regular monotone iterative scheme

an algebraic approach to polynomial reproduction of hermite subdivision schemes abstract we present an accurate investigation of the algebraic conditions that the symbols of a univariate binary hermite subdivision scheme have to fulfil in order to reproduce polynomials these conditions are sufficient for the scheme to satisfy the so called spectral condition the latter requires the existence of particular polynomial eigenvalues of the stationary counterpart of the hermite scheme in accordance with the known hermite schemes we here consider the case of a hermite scheme dealing with function values first and second derivatives several examples of application of the proposed algebraic conditions are given in both the primal and the dual situation

a decomposition of solomon s descent algebra abstract a descent class in the symmetric group sn is the collection of permutations with a given descent set it was shown by l solomon j algebra41 1976 255 264 that the product in the group algebra q sn of two descent classes is a linear combination of descent classes thus descent classes generate a subalgebra of q sn we refer to it here as solomon s descent algebra and denote it by n this algebra is not semisimple but it has a faithul representation in terms of upper triangular matrices the main goal of this paper is a decomposition of its multiplicative structure it develops that n acts in a natural way on the so called lie monomials this action has a purely combinatorial description and is a crucial tool in the construction of a complete set of indecomposable representations of n in particular we obtain a natural basis of irreducible orthogonal idempotents indexed by partitions of n for the quotient n n natural bases of nilpotents and idempotents for the subspaces e ne for two arbitrary partitions and are also constructed and the dimensions of these spaces are given a combinatorial interpretation in terms of the so called decreasing factorization of an arbitary word into a product of lyndon words

on lie powers of the augmentation ideal in characteristic 2 abstract let g be a finite group k a field and v a finite dimensional kg module write l v for the free lie algebra on v the action of g on v extends naturally to l v so it becomes a kg module which is direct sum of finite dimensional submodules for a finite dimensional kg module v let i v denote the set of isomorphism classes of indecomposable direct summands of v and i l v n 1 i l n v it is natural to ask whether i l v is finite of course this is not a question unless there exist infinitely many isomorphism classes of indecomposable kg modules that is k has positive characteristic p and the sylow p subgroups of g are non cyclic and dim v 2 let k be a field of characteristic 2 and g the klein four group we write d k g for the augmentation ideal of kg for the dual of and v 1 v 2 v 3 for the three 2 dimensional induced modules from the three cyclic subgroups of g in this article we concentrate on lie powers l n of and describe the module decomposition of l 16 a module of dimension 2690010 in particular i l 16 d v 1 v 2 v 3

modular lie powers and the solomon descent algebra let v be an r dimensional vector space over an infinite field f of prime characteristic p and let l n v denote the nth homogeneous component of the free lie algebra on v we study the structure of l n v as a module for the general linear group gl r f when n pk and k is not divisible by p and where r n our main result is an explicit 1 1 correspondence multiplicity preserving between the indecomposable direct summands of l k v and the indecomposable direct summands of l n v which are not isomorphic to direct summands of v n our approach uses idempotents of the solomon descent algebras and in addition a correspondence theorem for permutation modules of symmetric groups

monotone methods for a boundary value problem of second order discrete equation abstract the comparison and existence results of the upper and lower solutions are established for a boundary value problem of second order discrete equation a monotone iterative scheme is proposed the local as well as global existence and uniqueness of the solution are obtained the global convergence of the iteration is investigated the convergence rate of the iteration is analyzed and compared some examples are given

ellipse preserving hermite interpolation and subdivision we introduce a family of piecewise exponential functions that have the hermite interpolation property our design is motivated by the search for an effective scheme for the joint interpolation of points and associated tangents on a curve with the ability to perfectly reproduce ellipses we prove that the proposed hermite functions form a riesz basis and that they reproduce prescribed exponential polynomials we present a method based on green s functions to unravel their multi resolution and approximation theoretic properties finally we derive the corresponding vector and scalar subdivision schemes which lend themselves to a fast implementation the proposed vector scheme is interpolatory and level dependent but its asymptotic behavioris the same as the classical cubic hermite spline algorithm the same convergence properties i e fourth order of approximation are hence ensured

monotone iterative methods for nonlinear equations involving a noninvertible linear part we study monotone iterative methods for the numerical solution of a class of nonlinear elliptic boundary value problems by applying a theorem of ortega and rheinboldt this generalizes the work of earlier authors to the case of nonlinear perturbations of linear problems with a nontrivial kernel

a generalized taylor factorization for hermite subdivision schemes in a recent paper we investigated factorization properties of hermite subdivision schemes by means of the so called taylor factorization this decomposition is based on a spectral condition which is satisfied for example by all interpolatory hermite schemes nevertheless there exist examples of hermite schemes especially some based on cardinal splines which fail the spectral condition for these schemes and others we provide the concept of a generalized taylor factorization and show how it can be used to obtain convergence criteria for the hermite scheme by means of factorization and contractivity

modular lie representations of groups of prime order let k be a field of prime characteristic p and let g be a group of order p for any finite dimensional kg module v and any positive integer n let ln v denote the nth homogeneous component of the free lie k algebra generated by a basis of v then ln v can be considered as a kg module called the nth lie power of v the main result of the paper is a formula which describes the module structure of ln v up to isomorphism

some remarks on tilting modules double centralisers and lie modules abstract in a recent paper 14 s doty and j m douglass consider the degree r component l r e of the free lie algebra l e on an n dimensional vector space e over a suitable field k of characteristic not dividing r the authors discuss amongst other things the question of when this space regarded as a module for the general linear group of degree n equivalently the schur algebra s n r has the double centraliser property some general classes are described but the question in general is left open here we show that at most one block component of l r e can fail to have the double centraliser property and give some examples of the failure of the property for that block and therefore for the lie module

monotone iterative methods for finite difference system of reaction diffusion equations this paper presents an existence comparison theorem and an iterative method for a nonlinear finite difference system which corresponds to a class of semilinear parabolic and elliptic boundary value problems the basic idea of the iterative method for the computation of numerical solutions is the monotone approach which involves the notion of upper and lower solutions and the construction of monotone sequences from a suitable linear discrete system using upper and lower solutions as two distinct initial iterations two monotone sequences from a suitable linear system are constructed it is shown that these two sequences converge monotonically from above and below respectively to a unique solution of the nonlinear discrete equations this formulation leads to a well posed problem for the nonlinear discrete system applications are given to several models arising from physical chemical and biological systems numerical results are given to some of these models including a discussion on the rate of convergence of the monotone sequences

construction of hermite subdivision schemes reproducing polynomials abstract the aim of this study is to present a new class of quasi interpolatory hermite subdivision schemes of order two with tension parameters this class extends and unifies some of well known hermite subdivision schemes including the interpolatory hermite schemes acting on a function and the associated first derivative values each scheme in this class reproduces polynomials up to a certain degree depending on the size of stencil this is desirable property since the reproduction of polynomials up to degree d leads to the approximation order d 1 the smoothness analysis has been performed by using the factorization framework of subdivision operators lastly we present some numerical examples to demonstrate the performance of the proposed hermite schemes