center manifolds and contractions on a scale of banach spaces we give a new proof for the existence of a ck center manifold at a nonhyperbolic equilibrium point of a finite dimensional vector field of class ck the problem is reduced to a fixed point problem on a scale of banach spaces these banach spaces consist of mappings with a certain maximal exponential growth at infinity we give conditions under which there is a unique fixed point depending differentiably on the parameters the main difficulty is that the mappings under consideration become only differentiable after composition with appropriate embeddings on the scale of banach spaces

on the geometry of the term structure of interest rates we present recently developed geometric methods for the analysis of finite dimensional term structure models of the interest rates this includes an extension of the frobenius theorem for fr echet spaces in particular this approach puts new light on many of the classical models such as the hull white extended vasicek and cox ingersoll ross short rate models the notion of a finite dimensional realization fdr is central for this analysis we motivate it classify all generic fdrs and provide some new results for the corresponding factor processes such as hypoellipticity of its generators and the existence of smooth densities furthermore we include finite dimensional external factors thus admitting a stochastic volatility structure

on local rigidity of partially hyperbolic affine ℤ k actions the following dichotomy for affine z k actions on the torus t d k d is an element of n is shown to hold i the linear part of the action has no rank one factors and then the affine action is

a center manifold for second order semilinear differential equations on the real line and applications to the existence of wave trains for the gurtin mccamy equation

accurate solution of polynomial equations using macaulay resultant matrices we propose an algorithm for solving two polynomial equations in two variables our algorithm is based on the macaulay resultant approach combined with new techniques including randomization to make the algorithm accurate in the presence of roundoff error the ultimate computation is the solution of a generalized eigenvalue problem via the qz method we analyze the error due to roundoff of the method showing that with high probability the roots are computed accurately assuming that the input data that is the two polynomials are well conditioned our analysis requires a novel combination of algebraic and numerical techniques

generalized insertion region guides for delaunay mesh refinement mesh generation by delaunay refinement is a widely used technique for constructing guaranteed quality triangular and tetrahedral meshes the quality guarantees are usually provided in terms of the bounds on circumradius to shortest edge ratio and on the grading of the resulting mesh traditionally circumcenters of skinny elements and middle points of boundary faces and edges are used for the positions of inserted points however recently variations of the traditional algorithms are being proposed that are designed to achieve certain optimization objectives by inserting new points in neighborhoods of the center points in this paper we propose a general approach to the selection of point positions by defining one two and three dimensional selection regions such that any point insertion strategy based on these regions is automatically endowed with the theoretical guarantees proven here in particular for the input models defined by planar linear complexes under the assumption that no input angle is le

on the numerical solution of differential algebraic equations by pade series in this paper the solution for a differential algebraic equation dae can be expanded up to arbitrary order using maple computer algebra systems frank maple v crc press boca raton fl 1996 first we calculate power series of the given equations system then transform it into pade series form which gives an arbitrary order for solving dae numerically

generating well shaped d dimensional delaunay meshes a d dimensional simplicial mesh is a delaunay triangulation if the circumsphere of each of its simplices does not contain any vertices inside a mesh is well shaped if the maximum aspect ratio of all its simplices is bounded from above by a constant it is a long term open problem to generate well shaped d dimensional delaunay meshes for a given polyhedral domain in this paper we present a refinement based method that generates well shaped d dimensional delaunay meshes for any plc domain with no small input angles furthermore we show that the generated well shaped mesh has o n d simplices where n is the smallest number of d simplices of any almost good meshes for the same domain here a mesh is almost good if each of its simplices has a bounded circumradius to the shortest edge length ratio

tetrahedral mesh generation using delaunay refinement with non standard quality measures this paper studies the practical performance of delaunay refinement tetrahedral mesh generation algorithms by using non standard quality measures to drive refinement we show that sliver tetrahedra can be eliminated from constrained delaunay tetrahedralizations solely by refinement despite the fact that quality guarantees cannot be proven the algorithm can consistently generate meshes with dihedral angles between 18circ and 154 using a fairer quality measure targeting every type of bad tetrahedron dihedral angles between 14 and 154 can be obtained the number of vertices inserted to achieve quality meshes is comparable to that needed when driving refinement with the standard circumradius to shortest edge ratio we also study the use of mesh improvement techniques on delaunay refined meshes and observe that the minimum dihedral angle can generally be pushed above 20 regardless of the quality measure used to drive refinement the algorithm presented in this paper can accept geometric domains whose boundaries are piecewise smooth copyright 2011 john wiley sons ltd

hopf bifurcation with constraints in this study we give conditions for the existence of periodic orbits of a strongly continuous semiflow which branch off from a steady state for a certain parameter value the semiflow is defined in a closed subset of a banach space which contains the steady state the steady state is kept fixed whereas both the semiflow and its domain may depend strongly continuously on the parameter it will be shown that an asymptotic contraction hypothesis together with a transversality and no resonance hypothesis for the eigenvalues of the infinitesimal generator of the linearized semiflow imply hopf bifurcation the technique consists in a generalized lyapunov schmidt decomposition method using spectral projections which depend strongly continuously on the bifurcation parameter the existence of a continuum of bifurcating periodic orbits follows from a degree argument the theory can be applied to partial functional differential equations when the bifurcation parameter changes transformed arguments or the initial or boundary values underly parameter dependent constraints in general such semiflows do not exhibit the smoothness hypotheses which are necessary for the application of available hopf bifurcation theorems there are a lot of such theorems either for special types of functional differential equations partial differential equations volterra integral equations or for a general semiflow in banach space but under considerable stronger hypotheses than in the present paper the reader is referred to l 2 4 5 7 12 15 and to the references therein of course the stronger hypotheses in these papers allow to get more detailed information on the structure of the branch of the bifurcating periodic orbits such as smootheness with respect to the parameter local stability and the direction of bifurcation in contrast to them the hypotheses in our paper guarantee only local existence and they are too weak to ensure smoothness of the branch nor the existence of a center manifold but even the local existence problem was still unsolved if the semiflow and its domain depend merely strongly continuously on the parameter the difficulty is due to the fact that the usual argument basing on the implicit function theorem will not work some examples which demonstrate the application of our theory will be given in a subsequent paper before we will explain the main hypotheses we shall introduce some notations b denotes a real banach space and c c iw a compact neighborhood of 0 e r the domain of the bifurcation parameter norms and the modulus of a complex number are always indicated by the symbol 1 1 to every c e c we join a closed subset x c b the domain of the semiflow

hopf bifurcation for functional and functional differential equations with infinite delay here x is a vector in r the parameter c is an element of a finite dimensional real banach space d and f and g are mappings from d x buc r r into r moreover f a 0 g a 0 0 so x e 0 is a solution of both 1 1 and 1 2 in addition we suppose that the linearizations of 1 1 and 1 2 have a one parameter family of nontrivial periodic solutions at a critical value a0 of the parameter our aim is to show that the nonlinear equations have one parameter families of nontrivial periodic solutions for values of a close to ao most of the existing hopf bifurcation results for 1 1 and 1 2 require f and g to be nonanticipatory functionals with a finite delay i e the terms f a x and g a x are allowed to depend only on the values of x s for t r s t where r is some fixed finite number see e g l 4 5 13 14 261 some resent results do exists see 6 and 30 where f is allowed to include an infinite delay but then it is still assumed that f and g have an exponentially fading memory i e that f a x and g a x are well defined for functions x which grow exponentially at minus infinity in 9 a hopf bifurcation result is proved for a certain quadratic 114 oo22 0396187 3 00

the ordinary successive approximations method and pade approximants for solving a differential equation with variant retarded argument the aim of this paper is to present an efficient numerical procedure for solving boundary value problems for a differential equation with retarded argument x t a t x t t t f t x t f t l 0 0 0

geometric separators for finite element meshes we propose a class of graphs that would occur naturally in finite element and finite difference problems and we prove a bound on separators for this class of graphs graphs in this class are embedded in d dimensional space in a certain manner for d dimensional graphs our separator bound is o n d 1 d which is the best possible bound we also propose a simple randomized algorithm to find this separator in o n time this separator algorithm can be used to partition the mesh among processors of a parallel computer and can also be used for the nested dissection sparse elimination algorithm

recent developments in inverse acoustic scattering theory we survey some of the highlights of inverse scattering theory as it has developed over the last 15 years with emphasis on uniqueness theorems and reconstruction algorithms for time harmonic acoustic waves included in our presentation are numerical experiments using real data and numerical examples of the use of inverse scattering methods to detect buried objects

the lie algebra of homeomorphisms of the circle we define and study an infinite dimensional lie algebrahomeo which is shown to be naturally associated to thetopologicallie grouphomeo of all orientation preserving homeomorphisms of the circle roughly we rely on the universal decorated teichmuller theory developed before as motivation to provide frechet coordinates on the homogeneous space given byhomeo modulo the group of real fractional linear transformations whose corresponding vector fields on the circle we then extend by the usual lie algebrasl2of real traceless two by two matrices in order to definehomeo surprisingly homeo turns out to be equal to the algebra of all vector fields on the circle which are piecewisesl2 in the obvious sense it is evidently important to consider the relationship between our new frechet coordinates and the usual trigonometric functions on the circle and we undertake here both natural infinitesimal calculations we finally apply some further previous work in order to give sufficient conditions on the fourier coefficients of a certain class of homeomorphisms of the circle which arises naturally in topology and number theory

metric aspects of noncommutative homogeneous spaces for a closed cocompact subgroup of a locally compact group g given a compact abelian subgroup k of g and a homomorphism k g satisfying certain conditions landstad and raeburn constructed equivariant noncommutative deformations c g of the homogeneous space g generalizing rieffel s construction of quantum heisenberg manifolds we show that when g is a lie group and g is connected given any norm on the lie algebra of g the seminorm on c g induced by the derivation map of the canonical g action defines a compact quantum metric furthermore it is shown that this compact quantum metric space depends on continuously with respect to quantum gromov hausdorff distances

adaptive tetrahedral mesh generation by constrained delaunay refinement this paper presents a tetrahedral mesh generation method for numerically solving partial differential equations using finite element or finite volume methods in three dimensional space the main issues are the mesh quality and mesh size which directly affect the accuracy of the numerical solution and the computational cost two basic problems need to be resolved namely boundary conformity and field points distribution the proposed method utilizes a special three dimensional triangulation so called constrained delaunay tetrahedraliwtion to conform the domain boundary and create field points simultaneously good quality tetrahedra and graded mesh size can be theoretically guaranteed for a large class of mesh domains in addition an isotropic size field associated with the numerical solution can be supplied the field points will then be distributed according to it good mesh size conformity can be achieved for smooth sizing informations the proposed method has been implemented various examples are provided to illustrate its theoretical aspects as well as practical performance

quality mesh generation in higher dimensions we consider the problem of triangulating a d dimensional region our mesh generation algorithm called qmg is a quadtree based algorithm that can triangulate any polyhedral region including nonconvex regions with holes furthermore our algorithm guarantees a bounded aspect ratio triangulation provided that the input domain itself has no sharp angles finally our algorithm is guaranteed never to overrefine the domain in the sense that the number of simplices produced by qmg is bounded above by a factor times the number produced by any competing algorithm where the factor depends on the aspect ratio bound satisfied by the competing algorithm the qmg algorithm has been implemented in c and is used as a mesh generator for the finite element method

invariant manifolds for volterra integral equations of convolution type abstract in this paper we develop some elements of a qualitative theory for nonlinear volterra integral equations of convolution type our starting point is a local semiflow associated with the equation and acting on a space of compactly supported forcing functions within that framework we discuss the variation of constants formula the saddle point property the center manifold and hopf bifurcation some equations from population biology get special attention

backward error analysis for numerical integrators backward error analysis has become an important tool for understanding the long time behavior of numerical integration methods this is true in particular for the integration of hamiltonian systems where backward error analysis can be used to show that a symplectic method will conserve energy over exponentially long periods of time such results are typically based on two aspects of backward error analysis i it can be shown that the modified vector fields have some qualitative properties which they share with the given problem and ii an estimate is given for the difference between the best interpolating vector field and the numerical method these aspects have been investigated recently for example by benettin and giorgilli in j statist phys 74 1994 pp 1117 1143 by hairer in ann numer math 1 1994 pp 107 132 and by hairer and lubich in numer math 76 1997 pp 441 462 in this paper we aim at providing a unifying framework and a simplification of the existing results and corresponding proofs our approach to backward error analysis is based on a simple recursive definition of the modified vector fields that does not require explicit taylor series expansion of the numerical method and the corresponding flow maps as in the above cited works as an application we discuss the long time integration of chaotic hamiltonian systems and the approximation of time averages along numerically computed trajectories

stable manifolds in the method of averaging on considere l equation differentielle ef z t e ou f est t periodique en t et e 0 est un petit parametre et l equation moyennee f z i 1 t 0 t f z t o dt on suppose que l equation moyennee a un equilibre hyperbolique en z 0 a variete stable w soit e t la solution t periodique hyperbolique de ef z t e pres de z 0 on demontre un resultat sur la convergence reguliere de la variete stable de e t a w r quand e 0

endemic models with arbitrarily distributed periods of infection i fundamental properties of the model this paper is dedicated to frank hoppensteadt on the occasion of his 60th birthday abstract a model is developed for the spread of anin fectious disease ina populationwith constant recruitment of new susceptibles and the fundamental properties of its solutions are analyzed the model allows for arbitrarily many stages of infection all of which have general length distributions and disease mortalities existence and uniqueness of solutions to the model equations are established a basic reproduction ratio is derived and related to the existence of an endemic equilibrium to the stability of the disease free equilibrium and to weak and strong endemicity persistence of the disease a characteristic equation is found the zeros of which determine the local stability of the endemic equilibrium and sufficient stability conditions are given for the case that infected individuals do not return into the susceptible class in a subsequent paper explicit sufficient and necessary stability conditions will be derived for the case that the disease dynamics are much faster than the demographics

geodesics distance and the cat 0 property for the manifold of riemannian metrics given a fixed closed manifold m we exhibit an explicit formula for the distance function of the canonical l 2 riemannian metric on the manifold of all smooth riemannian metrics on m additionally we examine the metric completion of the manifold of metrics with respect to the l 2 metric and show that there exists a unique minimal path between any two points this path is also given explicitly as an application of these formulas we show that the metric completion of the manifold of metrics is a cat 0 space

stable finite elements for problems with wild coefficients we consider solving an elliptic boundary value problem in the case that the coefficients vary by many orders of magnitude over the domain a linear finite element method is used it is shown that the standard method for solving the resulting linear equations in finite precision arithmetic can give an arbitrarily inaccurate answer because of ill conditioning in the stiffness matrix a new method for solving the linear equations is proposed this method is based on a mixed formulation and gives a numerically accurate answer independent of the variation in the coefficients the numerical error in the soution of the linear system for the new method is shown to depend on the aspect ratio of the triangulation

bounded solutions to singularly perturbed systems of o d e abstract we consider existence and stability properties of bounded solutions to singular systems of o d e the method we suggest considers the concept of singular maps between scales of banach spaces and their properties this theory is then applied to obtain existence and transversality of bounded solutions to singular systems of o d e

on the regularity of harmonic functions and spherical harmonic maps defined on lattices abstract a growth lemma for certain discrete symmetric laplacians defined on a lattice z d zd rd with spacing is proved the lemma implies a de giorgi theorem that the harmonic functions for these laplacians are equi holder continuous 0 these results are then applied to establish regularity properties for the harmonic maps defined on zd and taking values in an n dimensional sphere sn uniform in questions of the convergence 0 and the dirichlet problem for these discrete harmonic maps are also addressed

percival variational principle for invariant measures and commensurate incommensurate phase transitions in one dimensional chains we prove for a one dimensional system of classical particles with potential energy u alpha gamma sum limits n left alpha v x n f x n 1 x n gamma right the existence of such a smooth function 0 0 that the system with potential energyu has the equilibrium state at the temperaturet 0 this is the incommensurate phase with the ratio of periods equal to the prescribed irrational number badly approximated by rational ones a simple geometric condition for the invariant curve of the corresponding dynamical system is established under which it is the support of the invariant measure minimizing percival s energy functional

cubature on wiener space in infinite dimension we prove a stochastic taylor expansion for stochastic partial differential equations spdes and apply this result to obtain cubature methods i e high order weak approximation schemes for spdes

solving elliptic finite element systems in near linear time with support preconditioners we consider linear systems arising from the use of the finite element method for solving scalar linear elliptic problems our main result is that these linear systems which are symmetric and positive semidefinite are well approximated by symmetric diagonally dominant matrices our framework for defining matrix approximation is support theory significant graph theoretic work has already been developed in the support framework for preconditioners in the diagonally dominant case and in particular it is known that such systems can be solved with iterative methods in nearly linear time thus our approximation result implies that these graph theoretic techniques can also solve a class of finite element problems in nearly linear time we show that the support number bounds which control the number of iterations in the preconditioned iterative solver depend on mesh quality measures but not on the problem size or shape of the domain

second order averaging and chaos in quasiperiodically forced weakly nonlinear oscillators abstract using the averaging method and melnikov s technique we study the chaotic dynamics of a class of quasiperiodically forced weakly nonlinear oscillators averaging is carried out to second order and the existence of transverse homoclinic tori resulting in chaotic dynamics for quasiperiodically forced oscillators is detected we also give an example for a duffing type weakly nonlinear oscillator subjected to combined parametric and external forcing in the case of double resonance in which primary external resonance and subharmonic or superharmonic parametric resonance exist simultaneously

well posedness of the water waves equations 1 1 presentation of the problem the water waves problem for an ideal liquid consists of describing the motion of the free surface and the evolution of the velocity field of a layer of perfect incompressible irrotational fluid under the influence of gravity in this paper we restrict our attention to the case when the surface is a graph parameterized by a function x where t denotes the time variable and x xi xd rd the horizontal spatial variables the method developed here works equally well for any integer d 1 but the only physically relevant cases are of course d 1 and d 2 the layer of fluid is also delimited from below by a not necessarily flat bottom parameterized by a time independent function b x we denote by q t the fluid domain at time t the incompressibility of the fluids is expressed by 1 1 div v 0 in qu t 0

a numerical solution of the elasticity problem of settled of the wronkler ground with variable coefficients in this paper we have given numerical solution of the elasticity problem of settled on the wronkler ground with variable coefficient the approximation solution of boundary value problem which is pertinent to this has been converted to integral equations and then by using the successive approximation methods has been reached in addition to this the approximation solution of the problem was put into pade series form we applied these methods to an example which is the elasticity problem of unit length homogeny beam which is a special form of boundary value problem first we calculate the successive approximation of the given boundary value problem then transform it into pade series form which give an arbitrary order for solving differential equation numerically

anisotropic delaunay mesh generation anisotropic meshes are triangulations of a given domain in the plane or in higher dimensions with elements elongated along prescribed directions anisotropic triangulations are known to be well suited for interpolation of functions or solving pdes assuming that the anisotropic shape requirements for mesh elements are given through a metric field varying over the domain we propose a new approach to anisotropic mesh generation relying on the notion of anisotropic delaunay meshes an anisotropic delaunay mesh is defined as a mesh in which the star of each vertex v consists of simplices that are delaunay for the metric associated to vertex v this definition works in any dimension and allows us to define a simple refinement algorithm the algorithm takes as input a domain and a metric field and provides after completion an anisotropic mesh whose elements are sized and shaped according to the metric field

an extended advancing front technique for closed surfaces mesh generation an extended advancing front technique aft with shift operations and riemann metric named as shifting aft is presented for finite element mesh generation on 3d surfaces especially 3d closed surfaces riemann metric is used to govern the size and shape of the triangles in the parametric space the shift operators are employed to insert a floating space between real space and parametric space during the 2d parametric space mesh generation in the previous work of closed surface mesh generation the virtual boundaries are adopted when mapping the closed surfaces into 2d open parametric domains however it may cause the mesh quality worsening problem in order to overcome this problem the aft kernel is combined with the shift operator in this paper the shifting aft can generate high quality meshes and guarantee convergence in both open and closed surfaces for the shifting aft it is not necessary to introduce virtual boundaries while meshing a closed surface hence the boundary discretization procedure is largely simplified and moreover better shaped triangles will be generated because there are no additional interior constraints yielded by virtual boundaries comparing with direct methods the shifting aft avoids costly and unstable 3d geometrical computations in the real space some examples presented in this paper have demonstrated the advantages of shift aft in 3d surface mesh generation especially for the closed surfaces copyright 2007 john wiley sons ltd

existence of invariant manifolds for stochastic equations in infinite dimension we provide a frobenius type existence result for finite dimensional invariant submanifolds for stochastic equations in infinite dimension in the spirit of da prato and zabczyk 5 we recapture and make use of the convenient calculus on frechet spaces as developed by kriegl and michor 16 our main result is a weak version of the frobenius theorem on frechet spaces as an application we characterize all finite dimensional realizations for a stochastic equation which describes the evolution of the term structure of interest rates

reliability of perturbation theory in general relativity the relation between perturbation theory and exact solutions in general relativity is tackled by investigating the existence and properties of smooth one parameter families of solutions on the one hand the coefficients of the taylor expansion in the parameter of any given smooth family of solutions necessarily satisfy the hierarchy of perturbation equations on the other hand it is the converse question does any solution of the perturbation equations come from taylor expanding some family of exact solutions which is of importance for the mathematical justification of the use of perturbation theory this converse question is called the one of the reliability of perturbation theory using and completing recent results on the characteristic initial value problem the local reliability of perturbation theory for general relativity in vacuum is proven very generally this result is then generalized to the einstein yang mills equations and therefore in particular to the einstein maxwell ones