a variable stepsize jacobian free exponential integrator for simulating transport in heterogeneous porous media application to wood drying a jacobian free variable stepsize method is developed for the numerical integration of the large stiff systems of differential equations encountered when simulating transport in heterogeneous porous media our method utilises the exponential rosenbrock euler method which is explicit in nature and requires a matrix vector product involving the exponential of the jacobian matrix at each step of the integration process these products can be approximated using krylov subspace methods which permit a large integration stepsize to be utilised without having to precondition the iterations this means that our method is truly jacobian free the jacobian need never be formed or factored during the simulation we assess the performance of the new algorithm for simulating the drying of softwood numerical experiments conducted for both low and high temperature drying demonstrates that the new approach outperforms in terms of accuracy and efficiency existing simulation codes that utilise the backward euler method via a preconditioned newton krylov strategy

algebraicity of local holomorphisms between real algebraic submanifolds of complex spaces we prove that a germ of a holomorphic map f between c n and c n sending one real algebraic submanifold mc n into another m c n is algebraic provided mcontains no complex analytic discs and m is generic and minimal we also propose an algorithm for finding complex analytic discs in a real submanifold

the meshfree finite volume method with application to multi phase porous media models numerical methods form a cornerstone of the analysis and investigation of mathematical models for physical processes many classical numerical schemes rely on the application of strict meshing structures to generate accurate solutions which in some applications are an infeasible constraint within this paper we outline a new meshfree numerical scheme which we call the meshfree finite volume method mfvm the mfvm uses interpolants to approximate fluxes in a disjoint finite volume scheme allowing for the accurate solution of strong form pdes we present a derivation of the mfvm and give error bounds on the spatial and temporal approximations used within the scheme we present a wide variety of applications of the method showing key features and advantages over traditional meshed techniques we close with an application of the method to a non linear multi phase wood drying model showing the potential for solving numerically challenging problems

algebraicity of holomorphic mappings between real algebraic sets in cn 0 introduction 1 holomorphic nondegenera y of real analytic manifolds 1 1 preliminaries on real submanifolds of c n 1 2 holomorphic nondegeneracy and its propagation 1 3 the levi number and essential finiteness 1 4 holomorphic nondegeneracy of real algebraic sets 2 the segre sets of a real analytic cr submanifold 2 1 complexification of m involution and projections 2 2 definition of the segre sets of m at p0 2 3 homogeneous submanifolds of cr dimension 1 2 4 homogeneous submanifolds of arbitrary cr dimension 2 5 proof of theorem 2 2 1 3 algebraic properties of holomorphic mappings between real algebraic sets 3 1 a generalization of theorems 1 and 4 3 2 propagation of algebralcity 3 3 proof of theorem 3 1 2 3 4 proof of theorem 3 1 8 3 5 an example 3 6 proofs of theorems 1 through 4

a higher order local linearization method for solving ordinary differential equations the local linearization ll method for the integration of ordinary differential equations is an explicit one step method that has a number of suitable dynamical properties however a major drawback of the ll integrator is that its order of convergence is only two the present paper overcomes this limitation by introducing a new class of numerical integrators called the llt method that is based on the addition of a correction term to the ll approximation in this way an arbitrary order of convergence can be achieved while retaining the dynamic properties of the ll method in particular it is proved that the llt method reproduces correctly the phase portrait of a dynamical system near hyperbolic stationary points to the order of convergence the performance of the introduced method is further illustrated through computer simulations

efficient simulation of unsaturated flow using exponential time integration abstract we assess the performance of an exponential integrator for advancing stiff semidiscrete formulations of the unsaturated richards equation in time the scheme is of second order and explicit in nature but requires the action of the matrix function a a 1 e a i on a suitability defined vector v at each time step when the matrix a is large and sparse a v can be approximated by krylov subspace methods that require only matrix vector products with a we prove that despite the use of this approximation the scheme remains second order furthermore we provide a practical variable stepsize implementation of the integrator by deriving an estimate of the local error that requires only a single additional function evaluation numerical experiments performed on two dimensional test problems demonstrate that this implementation outperforms second order variable stepsize implementations of the backward differentiation formulae

rate of convergence of local linearization schemes for initial value problems there is a large variety of local linearization ll schemes for the numerical integration of initial value problems which differ with respect to the algorithm that is used in the numerical implementation of the local linear discretization however in contrast with the ll discretization the order of convergence of the ll schemes have not been studied so far in this paper a general result about that matter is given in addition a brief survey of the main implementations of the ll method is also presented

highest weight vectors of the moonshine module with non zero graded trace abstract we study the space of graded traces one point correlation functions of highest weight vectors primary fields in the moonshine module we define a family of highest weight vectors with weight congruent to two modulo four and explicitly compute their graded traces in particular when their weight is at least 18 these highest weight vectors have a non zero graded trace

infinitesimal cr automorphisms of real hypersurfaces let m be an analytic real hypersurface through the origin in c n 1 and let hol m denote the space of vector fields x re z m tangent to m in a neighborhood of the origin where z is a holomorphic vector field defined in a neighborhood of the origin the hypersurface m is holomorphically nondegenerate at the origin if there is no holomorphic vector field tangent to m in a neighborhood of the origin the main result of the paper is that hol m is finite dimensional if and only if m is holomorphically nondegenerate at the origin

singular and maximal radon transforms analysis and geometry part 2 geometric theory 8 curvature introduction 8 1 three notions of curvature 8 2 theorems 8 3 examples 9 curvature some details 9 1 the exponential representation 9 2 diffeomorphism invariance 9 3 curvature condition cy 9 4 two lemmas 9 5 double fibration formulation 10 equivalence of curvature conditions 10 1 invariant submanifolds and deficient lie algebras 10 2 vanishing jacobians 10 3 construction of invariant submanifolds

exponential integrators for large systems of differential equations we study the numerical integration of large stiff systems of differential equations by methods that use matrix vector products with the exponential or a related function of the jacobian for large problems these can be approximated by krylov subspace methods which typically converge faster than those for the solution of the linear systems arising in standard stiff integrators the exponential methods also offer favorable properties in the integration of differential equations whose jacobian has large imaginary eigenvalues we derive methods up to order 4 which are exact for linear constant coefficient equations the implementation of the methods is discussed numerical experiments with reaction diffusion problems and a time dependent schrodinger equation are included

on cr mappings between algebraic cauchy riemann manifolds and separate algebraicity for holomorphic functions we prove the algebraicity of smooth cr mappings between algebraic cauchy riemann manifolds a generalization of separate algebraicity principle is established

regularity of mappings into classical domains we study the regularity and algebraicity for mappings into classical domains among other things we establish everywhere regularity and algebraicity results for c 2 cr maps into the set of smooth boundary points of a classical domain where the codimension can be arbitrarily large

transversality of cr mappings we prove here new results about transversality and related geometric properties of a holomorphic formal or cr mapping sending one generic submanifold of c into another one of our main results is that a finite mapping is transversal to the target manifold provided this manifold is of finite type for the case of hypersurfaces transversality in this context was proved by baouendi and the second author in 1990 the general case of generic manifolds of higher codimension which we treat in this paper had remained an open problem since then applications of this result include a sufficient condition for a finite mapping to be a local diffeomorphism

new efficient substepping methods for exponential timestepping exponential integrators are time stepping schemes which exactly solve the linear part of a semilinear ode system this class of schemes requires the approximation of a matrix exponential in every step and one successful modern method is the krylov subspace projection method we investigate the effect of breaking down a single timestep into arbitrary multiple substeps recycling the krylov subspace to minimise costs for these recycling based schemes we analyse the local error investigate them numerically and show they can be applied to a large system with 106 unknowns we also propose a new second order integrator that is found using the extra information from the substeps to form a corrector to increase the overall order of the scheme this scheme is seen to compare favorably with other order two integrators

algorithms for solving richards equation for variably saturated soils previous work has shown that a one dimensional water content based finite difference algorithm can offer improved cpu efficiency in modeling infiltration into very dry heterogeneous soil when compared to similar pressure based algorithms the usefulness of the water content based method is limited however since it cannot be applied to saturated soils in this paper we develop two new methods which retain the advantages of water content based methods while still permitting fully saturated conditions in the first method we define a new variable for the transformed richards equation which has the characteristics of water content when soil is unsaturated and of pressure when soil is at or near saturation in addition to the transformed richards equation method an improved pressure based method which uses flux updating is presented both methods are implemented in algorithms using a preconditioned conjugate gradient method equation solver the performance of the algorithms is compared to that of an algorithm based on a mixed form of richards equation using modified picard iteration two test cases are examined the first test case examines two dimensional infiltration into very dry heterogeneous soil which remains unsaturated throughout the simulation period the second test case examines a two dimensional developing perched water table results indicate that the new methods retain the advantages of the water content based method for fully unsaturated heterogeneous problems while allowing for fully saturated conditions for the unsaturated problem the new algorithms were from 16 to 59 times faster than the algorithm based on the mixed form of richards equation using modified picard iteration for the saturated problem the new algorithms were from 6 to 25 times faster than the mixed base algorithm results also indicate that the performance of the mixed base algorithm could be significantly improved if a more effective adaptive time step were developed

analysis of some krylov subspace approximations to the matrix exponential operator in this note a theoretical analysis of some krylov subspace approximations to the matrix exponential operation exp a v is presented and a priori and a posteriors error estimates are established several such approximations are considered the main idea of these techniquesis to approximately project the exponential operator onto a small krylov subspace and to carry out the resulting small exponential matrix computation accurately this general approach which has been used with success in several applications provides a systematic way of defining high order explicit type schemes for solving systems of ordinary differential equations or time dependent partial differential equations

dynamic properties of the local linearization method for initial value problems some dynamic properties of the local linearization ll scheme for the numerical integration of initial value problems in ordinary differential equations odes are investigated specifically the general conditions under which this scheme preserves the stationary points and periodic orbits of the odes and the local stability at these steady states are studied these dynamic properties are also examined by means of numerical experiments and the results are compared with those achieved by other numerical schemes in addition a brief review of the computational implementations of the ll scheme is also presented

two scale computational modelling of water flow in unsaturated soils containing irregular shaped inclusions the focus of this paper is two dimensional computational modelling of water flow in unsaturated soils consisting of weakly conductive disconnected inclusions embedded in a highly conductive connected matrix when the inclusions are small a two scale richards equation based model has been proposed in the literature taking the form of an equation with effective parameters governing the macroscopic flow coupled with a microscopic equation defined at each point in the macroscopic domain governing the flow in the inclusions this paper is devoted to a number of advances in the numerical implementation of this model namely by treating the micro scale as a two dimensional problem our solution approach based on a control volume finite element method can be applied to irregular inclusion geometries and if necessary modified to account for additional phenomena e g imposing the macroscopic gradient on the micro scale via a linear approximation of the macroscopic variable along the microscopic boundary this is achieved with the help of an exponential integrator for advancing the solution in time this time integration method completely avoids generation of the jacobian matrix of the system and hence eases the computation when solving the two scale model in a completely coupled manner numerical simulations are presented for a two dimensional infiltration problem

algebraicity of real analytic hypersurfaces with maximal rank let m be a connected strongly pseudoconvex real analytic hypersurface in c n 1 suppose that m p is of maximal rank for any p m it is shown that the following two statements are equivalent i there exists a certain point p 0 m such that m p 0 is cr isomorphic to a germ of a real algebraic hypersurface ii for any point p m m p is cr isomorphic to a germ of a real algebraic hypersurface

mappings of real algebraic hypersurfaces a holomorphic function h z defined in a neighborhood of a point z0 e cn is algebraic if there are polynomials qj z j 0 k not all identically zero such that qk z h z k qo z 0 a holomorphic mapping is algebraic if all its components are algebraic a smooth real hypersurface m in cn is algebraic if it is contained in the zero set of a nontrivial real valued polynomial in z and z we assume throughout this paper that n 1 in wi webster proved the following celebrated result if m and m are algebraic hypersurfaces in cn with nondegenerate levi forms and if h is a biholomorphism defined in an open neighborhood of m and mapping m into m then h is algebraic previously it had long been known that if m and m are open subsets of spheres then the components of h are in fact rational functions poincare p tanaka t in this paper we go a step further and give a complete characterization of algebraic hypersurfaces in cn such that any holomorphic mapping with nonvanishing jacobian determinant between two such hypersurfaces must be algebraic our main result is stated in theorem 1 below n by a germ at p0 of a holomorphic vector field in c we shall mean a

parametrization of local cr automorphisms by finite jets and applications the goal of this work is to study the structure of the group of all local realanalytic cr automorphisms of a germ of a real analytic hypersurface m p in c or equivalently of all local biholomorphisms h c p c p mapping the germ m p into itself we also deal with the more general situation of a realanalytic cr manifold of arbitrary codimension see 2 for every fixed point p m we refer to this group of local biholomorphisms as the stability group of m p and denote it by aut m p endowed with the topology of uniform convergence on compact neighbourhoods of p it becomes a topological group this natural inductive limit topology can be described in the following way a sequence of germs of biholomorphisms hj aut m p converges to another such germ h aut m p if there exists a neighbourhood u of p in c to which all mappings hj extend holomorphically and such that hj h uniformly on u this group arises in a variety of different circumstances one of the most important ones where it naturally appears is the biholomorphic equivalence problem which consists in deciding whether two given germs of real analytic hypersurfaces are biholomorphically equivalent the biholomorphic equivalence problem has been studied since the beginning of the last century when poincare 37 observed probably for the first time that two real hypersurfaces in c for n 2 are not necessarily locally equivalent via a holomorphic transformation though they are via a c transformation today the biholomorphic equivalence problem for strongly pseudoconvex or levi nondegenerate hypersurfaces is pretty well understood based on the works of e cartan 14 15 in dimension two and the results of tanaka 42 and chern moser 17 in arbitrary dimension thanks to the works initiated in the 1970s a number of remarkable properties of the stability group of strongly pseudoconvex real analytic hypersurfaces have been discovered see e g the surveys by vitushkin 45 huang 30 and baouendi ebenfelt and rothschild 5 for complete discussions on this matter these properties depend on whether the germ m p can be biholomorphically mapped onto a piece of the unit sphere if there exists such a biholomorphism one says that m p is spherical for instance if m p is not spherical the group aut m p is

extending proper holomorphic mappings of positive codimension in this paper we obtain results on holomorphic continuation of proper holomorphic mappings between pseudoconvex domains with real analytic boundaries in complex spaces of different dimensions equivalently we obtain results concerning the analyticity of cauchy riemann mappings between real analytic pseudoconvex hypersurfaces in complex spaces of different dimensions to begin with we recall the corresponding results for mappings of equidimensional domains let d and d be bounded pseudoconvex domains with smooth boundaries in c if the boundaries of d and d are strictly pseudoconvex or more generally of finite type in the sense of d angelo 15 then every proper holomorphic map of d onto d extends smoothly t o according to the results of bell and catlin 7 8 and diederich and forn ess 19 for mappings between strictly pseudoconvex domains this was proved by fefferman 26 and nirenberg webster and yang 37 if the boundaries of the pseudoconvex domains d d c c are real analytic then every proper holomorphic mapping of d onto d extends holomorphically to a neighborhood of 3 according to baouendi and rothschild 2 and diederich and forn ess 21 this reflection principle was first discovered by lewy 34 and pin6uk 38 for mappings between strictly pseudoconvex domains in the case of biholomorphic mappings between weakly pseudoconvex domains with realanalytic boundaries the result follows from the work of baouendi jacobowitz and treves 5 results in this direction were obtained in recent years by several authors see the papers 4 6 18 22 33 47 49 in this paper we are treating the case when the domains d and d have different dimensions to be specific we assume that d e c and d c c n are bounded pseudoconvex domains with real analytic boundaries and n n l in this situation a proper holomorphic map f d d need not be regular at the boundary for instance there exist proper holomorphic maps of balls of different dimensions

convergence and finite determination of formal cr mappings in this paper we study the convergence and finite determination of formal holomorphic mappings of c 0 taking one real submanifold into another by a formal holomorphic mapping h c 0 c 0 we mean an n vector h h1 hn where each hj is a formal power series in n indeterminates with no constant term if m and m are real smooth submanifolds through 0 in c defined near the origin by z z 0 and z z 0 respectively where and are vector valued smooth defining functions then we say that a formal mapping h c 0 c 0 sends m into m if the vector valued power series h z h z is a matrix multiple of z z for real analytic hypersurfaces we shall prove the following theorem 1 let m and m be real analytic hypersurfaces through the origin in c n 2 assume that neither m nor m contains a nontrivial holomorphic subvariety through 0 then any formal mapping h c 0 c 0 sending m into m is convergent the condition that m above does not contain a nontrivial holomorphic subvariety is necessary see remark 2 3 as a corollary we obtain the following characterization

determination of formal cr mappings by a finite jet abstract we prove that if m is a connected real analytic holomorphically nondegenerate hypersurface in c n 1 then for any point p m there exists an integer k such that any two germs at p of local biholomorphic mappings h 1 h 2 c n 1 p c n 1 p that send m into itself and whose k jets agree at p are identical the above is a special case of a more general theorem stated for formal hypersurfaces that gives a finite jet determination result for the class of formal mappings whose jacobian determinant does not vanish identically