a gradually implicit method with deferred correction and adaptive relaxation factors for simulation to nonlinear heat transfer abstract a gradually implicit method with deferred correction gimdc and adaptive relaxation factors arfs is developed for simulation to nonlinear heat transfer with temperature dependent properties gimdc is equivalent to explicit method initially and approaches implicit method gradually the theoretical analyses of convergence error and stability indicate that the accuracy and stability of gimdc are close to implicit method if arfs are utilized to guarantee convergence a strict condition for arfs is derived gimdc is compared with pem pure explicit method nim newton iterative method for implicit solution and fem finite element method which shows that gimdc is of acceptable numerical accuracy similar to nim and 19 28 and 12 89 higher than pem and fem respectively in average and higher efficiency the numerical tests are presented which shows that using optimized arfs confined by strict condition efficiency accuracy and stability are ensured

critical exponents of directed self avoiding walks it is argued that directed self avoiding random walks are in the same universality class as directed unrestricted random walks and therefore should exhibit mean field exponents gamma 1 nu sub 1 nu perpendicular to 1 2 in all dimensionalities

critical properties of directed self avoiding walks the generating functions and mean displacements of various two dimensional directed self avoiding walk models are calculated exactly by a simple transfer matrix approach asymptotically the authors find r sub n approximately rn2 1 2 approximately n and rperpendicular to n2 1 approximately n1 2 where n is the number of steps in the walk and and perpendicular to refer to projections of the displacement parallel and perpendicular to the preferred axis of the walk respectively some general properties of directed self avoiding walks for arbitrary dimensions are discussed as well

post lie algebra structures on pairs of lie algebras abstract we study post lie algebra structures on pairs of lie algebras g n which describe simply transitive nil affine actions of lie groups we prove existence results for such structures depending on the interplay of the algebraic structures of g and n we consider the classes of simple semisimple reductive perfect solvable nilpotent abelian and unimodular lie algebras furthermore we consider commutative post lie algebra structures on perfect lie algebras using lie algebra cohomology we can classify such structures in several cases we also study commutative structures on low dimensional lie algebras and on nilpotent lie algebras

renormalisation group study of fully directed self avoiding walks the author presents a discussion of the critical properties of fully directed self avoiding walks on hypercubic lattices a position space renormalisation group transformation is carried out analytically in two dimensions for any value of the scaling factor b the finite size scaling properties of the transformation are explicitly exhibited including the well known finite size scaling hypothesis v b v approximately 1n b 1 it is shown that the rg transformation in this case is sensitive to v sub thus giving support to existing arguments

polycyclic by finite groups admit a bounded degree polynomial structure for a polycyclic by finite group gamma of hirsch length h an affine resp polynomial structure is a representation of gamma into rm aff bbb r h resp rm p bbb r h the group of polynomial diffeomorphisms letting gamma act properly discontinuously on bbb r h recently it was shown by counter examples that there exist groups gamma even nilpotent ones which do not admit an affine structure thus giving a negative answer to a long standing question of john milnor we prove that every polycyclic by finite group gamma admits a polynomial structure which moreover appears to be of a special simple type called canonical and as a consequence of this consists entirely of polynomials of a bounded degree the construction of this polynomial structure is a special case of an iterated seifert fiber space construction which can be achieved here because of a very strong and surprising cohomology vanishing theorem

a short note on directed self avoiding walks the critical behaviour of partially directed self avoiding walks saws on a cubic lattice with only one axis directed is studied using exact enumerations the critical exponents nu sub 0 999 or 0 005 nu perpendicular to 0 501 or 0 006 and gamma 1 000 or 0 001 found here further support the universality hypothesis that various partially directed saws belong to the same universality class as fully directed saws and they are also dimension independent for d or 2

application of directed lattice animal theory to river networks it is shown that the theory of directed lattice animals is a good candidate for determining the exponents which describe river networks

a high order time splitting method based on integral deferred correction for semi lagrangian vlasov simulations semi lagrangian schemes with various splitting methods and with different reconstruction interpolation strategies have been applied to kinetic simulations for example the order of spatial accuracy of the algorithms proposed in qiu and christlieb 2010 29 is very high as high as ninth order however the temporal error is dominated by the operator splitting error which is second order for strang splitting it is therefore important to overcome such low order splitting error in order to have numerical algorithms that achieve higher orders of accuracy in both space and time in this paper we propose to use the integral deferred correction idc method to reduce the splitting error specifically the temporal order accuracy is increased by r with each correction loop in the idc framework where r 1 2r 1 2 for coupling the first order splitting and the strang splitting respectively the proposed algorithm is applied to the vlasov poisson system the guiding center model and two dimensional incompressible flow simulations in the vorticity stream function formulation we show numerically that the idc procedure can automatically increase the order of accuracy in time we also investigate numerical stability of the proposed algorithm via performing fourier analysis to a linear model problem

high order operator splitting methods based on an integral deferred correction framework integral deferred correction idc methods have been shown to be an efficient way to achieve arbitrary high order accuracy and possess good stability properties in this paper we construct high order operator splitting schemes using the idc procedure to solve initial value problems ivps we present analysis to show that the idc methods can correct for both the splitting and numerical errors lifting the order of accuracy by r with each correction where r is the order of accuracy of the method used to solve the correction equation we further apply this framework to solve partial differential equations pdes numerical examples in two dimensions of linear and nonlinear initial boundary value problems are presented to demonstrate the performance of the proposed idc approach

commutative post lie algebra structures and linear equations for nilpotent lie algebras abstract we show that for a given nilpotent lie algebra g with z g g g all commutative post lie algebra structures or cpa structures on g are complete this means that all left and all right multiplication operators in the algebra are nilpotent then we study cpa structures on free nilpotent lie algebras f g c and discover a strong relationship to solving systems of linear equations of type x u y v 0 for generator pairs x y f g c we use results of remeslennikov and stohr concerning these equations to prove that for certain g and c the free nilpotent lie algebra f g c has only central cpa structures

parisi s mean field solution for spin glasses as an analytic continuation in the replica number parisi s replica symmetry breaking solution for spin glasses is extended to finite replica number n the free energy fp n obtained this way as well as its first two derivatives with respect to n are shown to join the corresponding values in the sherrington kirkpatrick sk solution at a characteristic value ns t where stability breaks down in the latter the continuation composed of the sk branch fsk n for n or ns t and the parisi branch fp n for 0 or n or ns t fulfils the requirements of convexity monotonicity and stability for all n

on the zariski closure of the linear part of a properly discontinuous group of affine transformations let be a subgroup of the group of affine transformations of the affine space r 2n 1 suppose acts properly discontinuously on r 2n 1 the paper deals with the question which subgroups of gl 2n 1 r occur as zariski closure of the linear part of such a group the two main results of the paper say that so n 1 n does occur as of such a group if n is odd but does not if n is even

critical exponent of a directed self avoiding walk the authors have investigated numerically and analytically the directed self avoiding walk problem recently proposed and studied by chakrabarti and manna 1983 they find a different result for the exponent nu namely nu 1

flory theory for directed lattice animals and directed percolation the free energies of directed lattice animals in good and theta solvents and the free energy of directed percolation are found by the use of a simple flory type approximation which accounts for the inherent anisotropy of these systems from these free energies the authors obtain the upper critical dimension below which mean field theory breaks down they also calculate closed form dimension dependent expressions for the parallel and perpendicular correlation length exponents which characterise the asymptotic cluster shapes these exponents are in excellent agreement with existing numerical data

generalised self avoiding walk a generalisation of the self avoiding walk is introduced in which k or higher multiple points are forbidden k 2 corresponds to the standard self avoiding walk the flory theory gives the radius of gyration exponent nu k k 1 k 1 e 2 when e or ec k 2k k 1 e is the euclidean dimension of the problem and ec k the upper critical dimension which is also obtained using the fractal set theory

an arbitrary order fully implicit hybrid kinetic solver for linear radiative transport using integral deferred correction abstract in this work we describe the implementation of an arbitrarily high order hybrid solver for linear kinetic radiative transport equations the hybrid method is derived from a splitting of the radiative flux into free streaming and collisional components to which high and low resolution discrete ordinates methods are applied respectively arbitrarily high orders of accuracy with respect to time and space are attained by combining an integral deferred correction idc time integration scheme constructed with implicit euler substepping on radau ii right biased nodes with an upwind discontinuous galerkin dg spatial discretization on uniform cartesian meshes numerical experiments are used to demonstrate that the aforementioned idc methods can be constructed such that they are unconditionally stable l stable to within machine precision asymptotic analysis is used to show that such idc methods also preserve the diffusion limit of the underlying transport equation on the semi discrete level in the sense that the semi discrete transport system under an implicit idc discretization recovers the same idc discretization of the limiting diffusion equation convergence results in one spatial dimension are provided and it is found that while the hybrid method exhibits convergence stagnation and order reduction in certain scenarios the overall accuracy of the hybrid approximation is comparable to a standard discrete ordinates approximation in many cases numerical results for two test problems in two spatial dimensions are given to compare the computational efficiency of the hybrid method against a standard discrete ordinates method and to compare the efficiency of space time discretizations of different orders of accuracy the results indicate that a properly chosen hybrid discrete ordinates method can be more efficient than a standard discrete ordinates method by a factor of between 2 and 8 when idc timestepping methods are used further among the first third and fifth order space time discretizations considered here the first order method is the least efficient and the third order method is the most efficient

commutative post lie algebra structures on nilpotent lie algebras and poisson algebras abstract we give an explicit description of commutative post lie algebra structures on some classes of nilpotent lie algebras for non metabelian filiform nilpotent lie algebras and lie algebras of strictly upper triangular matrices we show that all cpa structures are associative and induce an associated poisson admissible algebra

an enhanced parareal algorithm based on the deferred correction methods for a stiff system in this study we consider a variant of the hybrid parareal algorithm based on deferred correction techniques in order to increase the convergence order even for the stiff system a hybrid parareal scheme introduced by minion 2011 20 improves the efficiency of the original parareal by utilizing a spectral deferred correction sdc strategy for a fine propagator within the parareal iterations in this paper we use krylov deferred correction kdc for a fine propagator to solve the stiff system and differential algebraic equations daes stably also we employ a deferred correction technique based on the backward euler method for a coarse propagator in order to make the global order of accuracy reasonably high while limiting the cost of sequential steps as small as possible numerical experiments on the efficiency of our method are promising

commutative post lie algebra structures on lie algebras abstract we show that any cpa structure commutative post lie algebra structure on a perfect lie algebra is trivial furthermore we give a general decomposition of inner cpa structures and classify all cpa structures on complete lie algebras as a special case we obtain the cpa structures of parabolic subalgebras of semisimple lie algebras

a zassenhaus conjecture and cpa structures on simple modular lie algebras abstract commutative post lie algebra structures on lie algebras in short cpa structures have been studied over fields of characteristic zero in particular for real and complex numbers motivated by geometry a perfect lie algebra in characteristic zero only admits the trivial cpa structure in this article we study these structures over fields of characteristic p 0 we show that every perfect modular lie algebra in characteristic p 2 having a solvable outer derivation algebra admits only the trivial cpa structure this involves a conjecture by hans zassenhaus saying that the outer derivation algebra out g of a simple modular lie algebra g is solvable we try to summarize the known results on the zassenhaus conjecture and prove some new results using the classification of simple modular lie algebras by premet and strade for algebraically closed fields of characteristic p 3 as a corollary we obtain that every central simple modular lie algebra of characteristic p 3 admits only the trivial cpa structure

the auslander conjecture for nil affine crystallographic groups let n be a simply connected connected real nilpotent lie group of finite dimension n we study subgroups in aff n n aut n acting properly discontinuously and cocompactly on n this situation is a natural generalization of the so called affine crystallographic groups we prove that for all dimensions 1 n 5 the generalized auslander conjecture holds i e that such subgroups are virtually polycyclic

on the apparent failure of a flory approximation for directed linear polymers the author discusses the apparent failure of a flory approximation for directed linear polymers commenting on the details implicit in this type of picture the relationship of a standard flory viewpoint to a simplified application of fractal theory is explored and a correspondence is drawn between aspects of both theories

conservative multi implicit spectral deferred correction methods for reacting gas dynamics in most models of reacting gas dynamics the characteristic time scales of chemical reactions are much shorter than the hydrodynamic and diffusive time scales rendering the reaction part of the model equations stiff moreover non linear forcings may introduce into the solutions sharp gradients or shocks the robust behavior and correct propagation of which require the use of specialized spatial discretization procedures this study presents high order conservative methods for the temporal integration of model equations of reacting flows by means of a method of lines discretization on the flux difference form of the equations these methods compute approximations to the cell averaged or finite volume solution the temporal discretization is based on a multi implicit generalization of spectral deferred correction methods the advection term is integrated explicitly and the diffusion and reaction terms are treated implicitly but independently with the splitting errors reduced via the spectral deferred correction procedure to reduce computational cost different time steps may be used to integrate processes with widely differing time scales numerical results show that the conservative nature of the methods allows a robust representation of discontinuities and sharp gradients the results also demonstrate the expected convergence rates for the methods of orders three four and five for smooth problems