the general properties of discrete time competitive dynamical systems abstract for strongly competitive discrete time dynamical systems on a strongly ordered topological vector space it is proved that any or limit set is unordered and lies on some invariant hypersurface with codimension one which generalizes hirsch s results for competitive autonomous systems of ordinary differential equations to competitive maps in a very general framework and implies the sarkovskii s theorem for planar strongly competitive and monotone maps we also show that the definition of competitiveness gives more restriction on homeomorphic property

construction of wavelets and prewavelets over triangulations constructions of wavelets and prewavelets over triangulations with an emphasis of the continuous piecewise polynomial setting are discussed some recent results on piecewise linear prewavelets and orthogonal wavelets are presented

the frequency decomposition multi grid method a new variant of the multi grid algorithms is presented it uses multiple coarse grid corrections with particularly associated prolongations and restrictions in this paper the robustness with respect to anisotropic problems is considered

stable three point wavelet bases on general meshes recently we introduced a wavelet basis on general possibly locally refined linear finite element spaces each wavelet is a linear combination of three nodal basis functions independently of the number of space dimensions in the present paper we show h s stability of this basis for a range of s that in any case includes s 1 which means that the corresponding additive schwarz preconditioner is optimal for second order problems furthermore we generalize the construction of the wavelet basis to manifolds we show that the wavelets have at least one and in areas where the manifold is smooth and the mesh is uniform even two vanishing moments because of these vanishing moments apart from preconditioning the basis can be used for compression purposes for a class of integral equation problems the stiffness matrix with respect to the wavelet basis will be close to a sparse one in the sense that a priori it can be compressed to a sparse matrix without the order of convergence being reduced

swapping edges of arbitrary triangulations to achieve the optimal order of approximation in the representation of scattered data by smooth pp piecewise polynomial functions perhaps the most important problem is to find an optimal triangulation of the given sample sites called vertices of course the notion of optimality depends on the desirable properties in the approximation or modeling problems in this paper we are concerned with optimal approximation order with respect to the given order r of smoothness and degree k of the polynomial pieces of the smooth pp functions we will only consider c1 pp approximation with r 1 and k 4 the main result in this paper is an efficient method for triangulating any finitely many arbitrarily scattered sample sites such that these sample sites are the only vertices of the triangulation and that for any discrete data given at these sample sites there is a c1 piecewise quartic polynomial on this triangulation that interpolates the given data with the fifth order of approximation

locally supported rational spline wavelets on a sphere in this paper we construct certain continuous piecewise rational wavelets on arbitrary spherical triangulations giving explicit expressions of these wavelets our wavelets have small support a fact which is very important in working with large amounts of data since the algorithms for decomposition compression and reconstruction deal with sparse matrices we also give a quasi interpolant associated to a given triangulation and study the approximation error some numerical examples are given to illustrate the efficiency of our wavelets

finite element wavelets with improved quantitative properties in w dahmen r stevenson element by element construction of wavelets satisfying stability and moment conditions siam j numer anal 37 1 1999 319 352 electronic finite element wavelets were constructed on polygonal domains or lipschitz manifolds that are piecewise parametrized by mappings with constant jacobian determinants the wavelets could be arranged to have any desired order of cancellation properties and they generated stable bases for the sobolev spaces h s for s 32 or s 1 on manifolds unfortunately it appears that the quantitative properties of these wavelets are rather disappointing in this paper we modify the construction from the above mentioned work to obtain finite element wavelets which are much better conditioned

cech cohomology of attractors of discrete dynamical systems abstract let f r n r n be a homeomorphism and k an asymptotically stable attractor for f the aim of this paper is to study when the inclusion of k in its basin of attraction a k induces isomorphisms in cech cohomology we show that i this is true if coefficients are taken in q or z p p prime and ii it is true for integral cohomology if and only if the cech cohomology of k or a k is finitely generated we compute the cech cohomology of periodic point free attractors of volume contracting r 3 homeomorphisms and present applications to quite general models in population dynamics

robust solution of singularly perturbed problems using multigrid methods we consider the problem of solving linear systems of equations that arise in the numerical solution of singularly perturbed ordinary and partial differential equations of reaction diffusion type standard discretization techniques are not suitable for such problems and so specially tailored methods are required usually involving adapted or fitted meshes that resolve important features such as boundary and or interior layers in this study we consider classical finite difference schemes on the layer adapted meshes of shishkin and bakhvalov we show that standard direct solvers exhibit poor scaling behavior with respect to the perturbation parameter when solving the resulting linear systems we propose and prove optimality of a new block structured preconditioning approach that is robust for small values of the perturbation parameter and compares favorably with standard robust multigrid preconditioners for these linear systems we also derive stopping criteria which ensure that the potential accuracy

robustness of multi grid applied to anisotropic equations on convex domains and on domains with re entrant corners we analyse multi grid applied to anisotropic equations within the framework of smoothing and approximation properties developed by hack busch for a model anisotropic equation on a square we give an up till now missing proof of an estimate concerning the approximation property which is essential to show robustness furthermore we show a corresponding estimate for a model anisotropic equation on an l shaped domain the existing estimates for the smoothing property are not suitable to prove robustness for either 2 cyclic gauss seidel smoothers or for less regular problems such as our second model equation for both cases we give sharper estimates by combination of our results concerning smoothing and approximation properties robustness of w cycle multi grid applied to both our model equations will follow for a number of smoothers

linear independence and stability of piecewise linear prewavelets on arbitrary triangulations in this paper we establish linear independence and stability of certain piecewise linear prewavelets over arbitrary bounded triangulations these prewavelets are natural generalizations of the locally supported element constructed by kotyczka and oswald for an infinite three directional mesh

on two ways of stabilizing the hierarchical basis multilevel methods a survey of two approaches for stabilizing the hierarchical basis hb multilevel preconditioners both additive and multiplicative is presented the first approach is based on the algebraic extension of the two level methods exploiting recursive calls to coarser discretization levels these recursive calls can be viewed as inner iterations at a given discretization level exploiting the already defined preconditioner at coarser levels in a polynomially based inner iteration method the latter gives rise to hybrid type multilevel cycles this is the so called hybrid algebraic multilevel iteration amli method the second approach is based on a different direct multilevel splitting of the finite element discretization space this gives rise to the so called wavelet multilevel decomposition based on l 2 projections which in practice must be approximated both approaches the amli one and the one based on approximate wavelet decompositions lead to optimal relative condition numbers of the multilevel preconditioners

c 1 spline wavelets on triangulations in this paper we investigate spline wavelets on general triangulations in particular we are interested in c 1 wavelets generated from piecewise quadratic polynomials by using the powell sabin elements we set up a nested family of spaces of c 1 quadratic splines which are suitable for multiresolution analysis of besov spaces consequently we construct c 1 wavelet bases on general triangulations and give explicit expressions for the wavelets on the three direction mesh a general theory is developed so as to verify the global stability of these wavelets in besov spaces the wavelet bases constructed in this paper will be useful for numerical solutions of partial differential equations

filter bank algorithms for piecewise linear prewavelets on arbitrary triangulations abstract this paper studies algorithms for decomposition reconstruction and approximation based on piecewise linear prewavelets on bounded triangulations of arbitrary topology our key mathematical result is showing that the schur complement of the associated two scale matrix is symmetric positive definite and well conditioned numerical examples suggest that thresholding based on prewavelets yields a smaller approximation error than when based on the simple faber decomposition scheme

carrying simplices in nonautonomous and random competitive kolmogorov systems abstract the purpose of this paper is to investigate the asymptotic behavior of positive solutions of nonautonomous and random competitive kolmogorov systems via the skew product flows approach it is shown that there exists an unordered carrying simplex which attracts all nontrivial positive orbits of the skew product flow associated with a nonautonomous random competitive kolmogorov system

new convergence estimates for multilevel algorithms for finite element approximations new convergence estimates are established for some multilevel algorithms for finite element methods applied to elliptic problems with jump coefficients a uniform rate of convergence is derived if the coefficient has only one jump interface if the coefficient has multi jump interfaces which meet at only one interior point in the domain the convergence rate is bounded by 1 cj 1 where j is the number of levels and c is a constant independent of the jump

saddle point behavior for monotone semiflows and reaction diffusion models the saddle point behavior is established for monotone semiflows with weak bistability structure and then these results are applied to three reaction diffusion systems modeling man environment man epidemics single loop positive feedback and two species competition respectively

the frequency decomposition multilevel method a robust additive hierarchical basis preconditioner hackbusch s frequency decomposition multi level method is characterized by the application of three additional coarse grid corrections in parallel to the standard one each coarse grid correction was designed to damp errors from a different part of the frequency spectrum in this paper we introduce a cheap variant of this method partly based on semi coarsening which demands less recursive calls than the original version using the theory of the additive schwarz methods we will prove robustness of our method as a preconditioner applied to anisotropic equations

on li yorke pairs the li yorke definition of chaos proved its value for interval maps in this paper it is considered in the setting of general topological dynamics we adopt two opposite points of view on the one hand sufficient conditions for li yorke chaos in a topological dynamical system are given we solve a long standing open question by proving that positive entropy implies li yorke chaos on the other hand properties of dynamical systems without li yorke pairs are investigated in addition to having entropy 0 they are minimal when transitive and the property is stable under factor maps arbitrary products and inverse limits finally it is proved that minimal systems without li yorke pairs are disjoint from scattering systems

multilevel schwarz methods we consider the solution of the algebraic system of equations which result from the discretization of second order elliptic equations a class of multilevel algorithms are studied using the additive schwarz framework we establish that the condition number of the iteration operators are bounded independent of mesh sizes and the number of levels this is an improvement on dryja and widlund s result on a multilevel additive schwarz algorithm as well as bramble pasciak and xu s result on the bpx algorithm some multiplicative variants of the multilevel methods are also considered we establish that the energy norms of the corresponding iteration operators are bounded by a constant less than one which is independent of the number of levels for a proper ordering the iteration operators correspond to the error propagation operators of certain v cycle multigrid methods using gauss seidel and damped jacobi methods as smoothers respectively

general explicit difference formulas for numerical differentiation through introducing the generalized vandermonde determinant the linear algebraic system of a kind of vandermonde equations is solved analytically by use of the basic properties of this determinant and then we present general explicit finite difference formulas with arbitrary order accuracy for approximating first and higher derivatives which are applicable to unequally or equally spaced data comparing with other finite difference formulas the new explicit difference formulas have some important advantages basic computer algorithms for the new formulas are given and numerical results show that the new explicit difference formulas are quite effective for estimating first and higher derivatives of equally and unequally spaced data

a basic norm equivalence for the theory of multilevel methods subspace decompositions of finite element spaces based onl 2 like orthogonal projections play an important role for the construction and analysis of multigrid like iterative methods recently several authors have proved the equivalence of the associated discrete norms with theh 1 norm the present paper gives an elementary self contained derivation of this result which is based on the use ofk functionals known from the theory of interpolation spaces

simplified analytical expressions for numerical differentiation via cycle index based on the lagrange interpolation to the function f x 0 for arbitrarily chosen x 0 and logarithmic differentiation we give a simple approach to analytical expressions for numerical differentiation using cycle index a detailed analysis for the remainder is also included

dynamics of the periodic type k competitive kolmogorov systems abstract for time periodic dissipative and irreducible type k competitive kolmogorov systems it is proved that there is a canonically defined countable family f of unordered disjoint invariant sets with the property that for every persistent trajectory whose limit set is not a cycle there exists a unique trajectory in some element of f such that these two trajectories are asymptotic and the corresponding points in these two trajectories are k related

convergence to equilibrium on invariant d hypersurfaces for strongly increasing discrete time semigroups abstract convergence to equilibrium for every positive semi orbit of strongly increasing discrete time semigroups on strongly ordered spaces is proved the domain of the semigroup is assumed to be slightly more general than a closed order interval in a strongly ordered banach space which is continuously imbedded into a banach lattice the semigroup is assumed to be order compact and every positive semi orbit is assumed to be order bounded the crucial hypothesis is the ljapunov stability of all equilibria it is also proved that the set of equilibria is a simply ordered arc the key tools are invariant d hypersurfaces

some estimates for a weighted l 2 projection this paper is devoted to the error estimates for some weighted l2 projections nearly optimal estimates are obtained these estimates can be applied to the analysis of the usual multigrid method multilevel preconditioner and domain decomposition method for solving elliptic boundary problems whose coefficients have large jump discontinuities

manifolds of balance in planar ecological systems abstract in the classic 2 species lotka volterra competition model and more general competitive planar kolmogorov models there is a continuous curve called the carrying simplex that links all non zero steady states and attracts all non zero population densities this curve is where the opposing processes of population growth and decline balance in this paper we use stability analysis and index theory to show that such a curve also exists when the interactions between two species are more general such as co operative or predator prey provided that reasonable biologically motivated conditions hold for example both species experience intraspecific competition and all population densities remain bounded for all time we consider systems where there is at most one co existence steady state the balance manifold is formed of heteroclinic orbits and attracts all non zero population densities but unlike its competitive analogue the curve is no longer necessarily continuously differentiable

chaotic attractors in the four dimensional leslie gower competition model abstract we study the occurrence of the chaotic attractor in the four dimensional classical leslie gower competition model we find that chaos can be generated by a cascade of quasiperiod doubling bifurcations starting from a supercritical neimark sacker bifurcation of the positive fixed point in this model the chaotic attractor is contained in the three dimensional carrying simplex that is a globally attracting invariant manifold biologically the result implies that the invasion attempts by an invader into a trimorphic population under the leslie gower dynamics can lead to chaos

fast wavelet transforms and numerical algorithms i a class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices in contrast the methods presented here are based on the recently developed theory of wavelets and are applicable to all calderon zygmund and pseudo differential operators the algorithms of this paper require order o n or o n log n operations to apply an n n matrix to a vector depending on the particular operator and the version of the algorithm being used and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here

piecewise linear prewavelets on arbitrary triangulations this paper studies locally supported piecewise linear prewavelets on bounded triangulations of arbitrary topology it is shown that a concrete choice of prewavelets form a basis of the wavelet space when the degree of the vertices in the triangulation is not too high

on the abstract theory of additive and multiplicative schwarz algorithms in recent years it has been shown that many modern iterative algorithms multigrid schemes multilevel preconditioners domain decomposition methods etc for solving problems resulting from the discretization of pdes can be interpreted as additive jacobi like or multiplicative gauss seidel like subspace correction methods the key to their analysis is the study of certain metric properties of the underlying splitting of the discretization space v into a sum of subspaces v j and the splitting of the variational problem on v into auxiliary problems on these subspaces in this paper we propose a modification of the abstract convergence theory of the additive and multiplicative schwarz methods that makes the relation to traditional iteration methods more explicit the analysis of the additive and multiplicative schwarz iterations can be carried out in almost the same spirit as in the traditional block matrix situation making convergence proofs of multilevel and domain decomposition methods clearer or at least more classical in addition we present a new bound for the convergence rate of the appropriately scaled multiplicative schwarz method directly in terms of the condition number of the corresponding additive schwarz operator these results may be viewed as an appendix to the recent surveys x ys

smoothness of the carrying simplex for discrete time competitive dynamical systems a characterization of neat embedding abstract the paper is concerned with the question of smoothness of the carrying simplex s for a discrete time dissipative competitive dynamical system we give a necessary and sufficient criterion for s being a c 1 submanifold with corners neatly embedded in the nonnegative orthant formulated in terms of inequalities between lyapunov exponents for ergodic measures supported on the boundary of the orthant this completes one thread of investigation occasioned by a question posed by m w hirsch in 1988 besides amenable conditions are presented to guarantee the c r r 1 smoothness of s in the time periodic competitive kolmogorov systems of odes examples are also presented one in which s is of class c 1 but not neatly embedded the other in which s is not of class c 1

uniqueness and attractivity of the carrying simplex for discrete time competitive dynamical systems abstract for the discrete time dissipative competitive dynamical system on r n it is proved that generically there is a unique n 1 dim invariant carrying simplex attracting all nontrivial orbits which solved the conjecture raised by h l smith in 1986

the frequency decomposition multi grid method ii convergence analysis based on the additive schwarz method the convergence analysis of a special variant of the additive schwarz iteration is presented it can be applied to the frequency decomposition multigrid method and yields robust convergence

multilevel preconditioning this paper is concerned with multilevel techniques for preconditioning linear systems arising from galerkin methods for elliptic boundary value problems a general estimate is derived which is based on the characterization of besov spaces in terms of weighted sequence norms related to corresponding multilevel expansions the result brings out clearly how the various ingredients of a typical multilevel setting affect the growth rate of the condition numbers in particular our analysis indicates how to realize even uniformly bounded condition numbers for example the general results are used to show that the bramble pasciak xu preconditioner for piecewise linear finite elements gives rise to uniformly bounded condition numbers even when the refinements of the underlying triangulations are highly nonuniform furthermore they are applied to a general multivariate setting of refinable shift invariant spaces in particular covering those induced by various types of wavelets

a robust hierarchical basis preconditioner on general meshes in this paper we introduce a multi level direct sum space decomposition of general possibly locally refined linear or multi linear finite element spaces the resulting additive schwarz preconditioner is optimal for symmetric second order elliptic problems moreover it turns out to be robust with respect to coefficient jumps over edges in the coarsest mesh perturbations with positive zeroth order terms and after a further decomposition of the spaces also with respect to anisotropy along the grid lines important for an efficient implementation is that stable bases of the subspaces defining our decomposition consisting of functions having small supports can be easily constructed

an algorithm for the finite difference approximation of derivatives with arbitrary degree and order of accuracy in this paper we introduce an algorithm and a computer code for numerical differentiation of discrete functions the algorithm presented is suitable for calculating derivatives of any degree with any arbitrary order of accuracy over all the known function sampling points the algorithm introduced avoids the labour of preliminary differencing and is in fact more convenient than using the tabulated finite difference formulas in particular when the derivatives are required with high approximation accuracy moreover the given matlab computer code can be implemented to solve boundary value ordinary and partial differential equations with high numerical accuracy the numerical technique is based on the undetermined coefficient method in conjunction with taylor s expansion to avoid the difficulty of solving a system of linear equations an explicit closed form equation for the weighting coefficients is derived in terms of the elementary symmetric functions this is done by using an explicit closed formula for the vandermonde matrix inverse moreover the code is designed to give a unified approximation order throughout the given domain a numerical differentiation example is used to investigate the validity and feasibility of the algorithm and the code it is found that the method and the code work properly for any degree of derivative and any order of accuracy

an exclusion principle for periodic competitive systems in three dimensions an exclusion principle for periodic competitive systems in three dimensionst rafael ortega and antonio tineo departamento de matemzitica aplicada universidad de granada 18071 granada spain and departmento de matem iticas facultad de ciencias universidad de los andes 5101 m6rida venezuela received 29 february 1996 received in revised form 25 september 1996 received for publication 18 february 1997 key words and phrases logistic equation competing species problem periodic solutions exclusion principle 0 introduction in this paper we consider the following three dimensional periodic system u uifi t ul u2 u u i 0 1 0 and uif t u is locally lipschitz continuous in u we shall assume that the system is strongly competitive and dissipative and prove the following exclusion principle theorem 0 1 suppose that a f t ul 2

linearization and invariant manifolds on the carrying simplex for competitive maps abstract a result due to m w hirsch states that most competitive maps admit a carrying simplex i e an invariant hypersurface of codimension one which attracts all nontrivial orbits the common approach in the study of these maps is to focus on the dynamical behavior on the carrying simplex however this manifold is normally non smooth therefore not every tool coming from differential geometry can be applied in this paper we prove that the restriction of the map to the carrying simplex in a neighborhood of an interior fixed point is topologically conjugate to the restriction of the map to its pseudo unstable manifold by an invariant foliation this implies that the linearization techniques are applicable for studying the local dynamics of the interior fixed points on the carrying simplex we further construct the stable and unstable manifolds on the carrying simplex our results give partial responses to hirsch s problem regarding the smoothness of the carrying simplex we discuss some applications in classical models of population dynamics