the cauchy problem for indefinite improper affine spheres and their hessian equation abstract we give a conformal representation for indefinite improper affine spheres which solve the cauchy problem for their hessian equation as consequences we can characterize their geodesics and obtain a generalized symmetry principle then we classify the helicoidal indefinite improper affine spheres and find a new family with geodesically complete non flat affine metric moreover we present interesting examples with singular curves and isolated singularities

a numerical continuation method based on pade approximants a continuation algorithm is presented with a new predictor which is based on a rational representation of the solution path this algorithm belongs to the class of asymptotic numerical methods that connect perturbation techniques with a discretization principle without the use of a correction process several examples from shell buckling and from contact mechanics are analyzed to assess the efficiency and the reliability of the method

the codazzi equation for surfaces abstract in this paper we study the classical codazzi equation in space forms from an abstract point of view and use it as an analytic tool to derive global results for surfaces in different ambient spaces in particular we study the existence of holomorphic quadratic differentials the uniqueness of immersed spheres in geometric problems height estimates and the geometry and uniqueness of complete or properly embedded weingarten surfaces

bifurcation points and bifurcated branches by an asymptotic numerical method and pade approximants the aim of this work is to develop a reliable and fast algorithm to compute bifurcation points and bifurcated branches it is based upon the asymptotic numerical method anm and pade approximants the bifurcation point is detected by analysing the poles of pade approximants or by evaluating along the computed solution branch a bifurcation indicator well adapted to anm several examples are presented to assess the effectiveness of the proposed method that emanate from buckling problems of thin elastic shells especially problems involving large rotations are discussed copyright 2004 john wiley sons ltd

numerical comparisons of high order nonlinear solvers for the transient navier stokes equations based on homotopy and perturbation techniques efficient solvers for the unsteady navier stokes equations are presented a classic time stepping scheme is combined with high order nonlinear solvers coupling homotopy and a perturbation technique polynomial and rational representations are used to approximate the unknowns of the problem a pseudo residual criterion is proposed to improve the efficiency of the solvers the numerical example considered in this paper is the time periodic two dimensional flow around a circular cylinder comparisons with the classical first order newton raphson solver are performed numerical results reveal that a lower number of matrix factorization is needed with the proposed methods decreasing the computational effort

bifurcation of low reynolds number flows in symmetric channels the flowfields in two dimensional channels with discontinuous expansions are studied numerically to understand how the channel expansion ratio influences the symmetric and nonsymmetric solutions that are known to occur for improved confidence and understanding two distinct numerical techniques are used the general flowfield characteristics in both symmetric and asymmetric regimes are ascertained by a time marching finite difference procedure the flowfields and the bifurcation structure of the steady solutions of the navier stokes equations are determined independently using the finite element technique the two procedures are then compared both as to their predicted critical reynolds numbers and the resulting flowfield characteristics following this both numerical procedures are compared with experiments

affine complete locally convex hypersurfaces an open problem in affine geometry is whether an affine complete locally uniformly convex hypersurface in euclidean n 1 space is euclidean complete for n 2 in this paper we give the affirmative answer as an application it follows that an affine complete affine maximal surface in r 3 must be an elliptic paraboloid

nonoscillation of a class of neutral differential equations for the neutral differential equation x t x t n p t x t 0 where n is an odd number 0 r we give a complete classification of nonoscillatory solutions and obtain conditions for the existence of each type of nonoscillatory solutions

ribaucour type transformations for the hessian one equation abstract we extend the classical theory of ribaucour transformations to the family of improper affine maps and use it to obtain new solutions of the hessian one equation we prove that such transformations produce complete embedded ends of parabolic type and curves of singularities which generically are cuspidal edges moreover we show that these ends and curves of singularities do no intersect we apply ribaucour transformations to some helicoidal improper affine maps providing new 3 parameter families with an interesting geometry and a good behavior at infinity in particular we construct improper affine maps periodic in one variable with any even number of complete embedded ends

on the maximal number of exceptional values of gauss maps for various classes of surfaces the main goal of this paper is to reveal the geometric meaning of the maximal number of exceptional values of gauss maps for several classes of immersed surfaces in space forms for example complete minimal surfaces in the euclidean three space weakly complete improper affine spheres in the affine three space and weakly complete flat surfaces in the hyperbolic three space for this purpose we give an effective curvature bound for a specified conformal metric on an open riemann surface

improper affine maps we study a class of improper affine spheres with singularities called improper affine maps new examples of genus 0 and 1 are described and the classification of complete improper affine maps with at most two embedded ends will be given

flat surfaces in the hyperbolic 3 space in this paper we give a conformal representation of flat surfaces in the hyperbolic 3 space using the complex structure induced by its second fundamental form we also study some examples and the behaviour at infinity of complete flat ends

oscillation criteria for second order nonlinear neutral delay dynamic equations in this paper we will establish some oscillation criteria for the second order nonlinear neutral delay dynamic equation r t y t p t y t f t y t 0 on a time scale t here 0 is a quotient of odd positive integers with r t and p t real valued positive functions defined on t to the best of our knowledge nothing is known regarding the qualitative behavior of these equations on time scales so this paper initiates the study

time integration algorithm based on divergent series resummation for ordinary and partial differential equations borel s technique of divergent series resummation is transformed into a numerical code and used as a time integration scheme it is applied to the resolution of regular and singular problems arising in fluid mechanics its efficiency is compared to those of classical discretization schemes

new oscillation criteria for first order nonlinear neutral delay differential equations some new oscillation criteria of first order nonlinear neutral delay differential equations with variable coefficients are given our results extend and improve some of the well known previously results in the literature some examples are considered to illustrate our main results

singular improper affine spheres from a given lagrangian submanifold abstract given a lagrangian submanifold l of the affine symplectic 2n space one can canonically and uniquely define a center chord and a special improper affine sphere of dimension 2n both of whose sets of singularities contain l although these improper affine spheres ias always present other singularities away from l the off shell singularities studied in 8 they may also present singularities other than l which are arbitrarily close to l the so called singularities on shell these on shell singularities possess a hidden z 2 symmetry that is absent from the off shell singularities in this paper we study these canonical ias obtained from l and their on shell singularities in arbitrary even dimensions and classify all stable lagrangian legendrian singularities on shell that may occur for these ias when l is a curve or a lagrangian surface

automatic detection and branch switching methods for steady bifurcation in fluid mechanics this paper deals with the computation of steady bifurcations in the framework of 2d incompressible navier stokes flow we first propose a numerical method to accurately detect the critical reynolds number where this kind of bifurcation appears from this singular value we introduce a numerical tool to compute all the steady bifurcated branches all these algorithms are based on the asymptotic numerical method 1 2 the critical values are determined by using a bifurcation indicator 3 5 and the bifurcated branches are computed by using an augmented system which was first introduced in solid mechanics 4 6 several numerical examples from 2d navier stokes show the reliability and the efficiency of the proposed methods

the dirichlet problem for monge ampere equations in non convex domains and spacelike hypersurfaces of constant gauss curvature in this paper we extend the well known results on the existence and regularity of solutions of the dirichlet problem for monge ampere equations in a strictly convex domain to an arbitrary smooth bounded domain in r as well as in a general riemannian manifold we prove for the nondegenerate case that a sufficient and necessary condition for the classical solvability is the existence of a subsolution for the totally degenerate case we show that the solution is in c11 q if the given boundary data extends to a locally strictly convex c2 function ori q as an application we prove some existence results for spacelike hypersurfaces of constant gauss kronecker curvature in minkowski space spanning a prescribed boundary

new oscillatory criteria for higher order nonlinear neutral delay differential equation abstract in this paper we consider the higher order nonlinear neutral delay differential equation 1 1 x t p t x t n f t x t 0 where n is an odd number and n 3 0 p c t 0 r t t lim t t f c t 0 r r f t u is nondecreasing in u

the indefinite affine cauchy problem abstract we solve the geometric cauchy problem for the class of affine maximal surfaces with indefinite affine metric in the affine space r 3 that is we find all the surfaces of this class which contain a given regular curve of r 3 with prescribed affine normal and affine conormal along it we prove the problem is well posed when the initial data are non characteristic and show that uniqueness of the solution can fail at characteristic asymptotic directions as application we obtain some results about geodesics and symmetries of indefinite affine maximal surfaces

an extension of a theorem by k jorgens and a maximum principle at infinity for parabolic affine spheres where is a planar domain and f is in the usual holder space c2 without loss of generality we shall consider only locally convex solutions of 1 this equation arises in the context of an affine differential geometric problem as the equation of a parabolic affine sphere in short pa sphere in the unimodular affine real 3 space see c1 c2 cy and lsz contrary to the case of smooth bounded convex domains little is known about solutions of 1 when the domain is unbounded here we recall a famous result by k jorgens which asserts that any solution of 1 on r2 is a quadratic polynomial see j and we also mention a previous paper see fmm where the authors study solutions of 1 on the exterior of a planar domain that are regular at infinity since the underlying almost complex structure of 1 is integrable one expects pa spheres with their canonical conformal structure to be conveniently described in terms of meromorphic functions the reader will find in sect 2 a complex representation of pa spheres and particularly a complex description for the solutions of 1

even dimensional improper affine spheres abstract there are exactly two different types of bi dimensional improper affine spheres the non convex ones can be modeled by the center chord transform of a pair of planar curves while the convex ones can be modeled by a holomorphic map in this paper we show that both constructions can be generalized to arbitrary even dimensions the former class corresponds to the center chord transform of a pair of lagrangian submanifolds while the latter is related to special kahler manifolds furthermore we show that the improper affine spheres obtained in this way are solutions of certain exterior differential systems finally we also discuss the problem of realization of simple stable legendrian singularities as singularities of these improper affine spheres

high performance computations of steady state bifurcations in 3d incompressible fluid flows by asymptotic numerical method this paper presents a powerful numerical model that implements the asymptotic numerical method to compute 3d steady state incompressible fluid flow solutions this continuation algorithm enables to explore branches of steady state solutions stable or unstable to accurately determine any simple steady state bifurcation points and their emanating bifurcated branches the powerfulness of the model stands on an optimal step length continuation thanks to the combination of power series analysis in the framework of anm along with an efficient parallel implementation of the resulting algorithm on high performance computers the outcome of this approach is demonstrated throughout 3d incompressible fluid flows inside a sudden expansion channel expansion ratio e 3 cross section aspect ratio 10 b 20 we have computed for the first time up to four steady symmetry breaking pitchfork bifurcations together with their associated bifurcated branches the main characteristic of this 3d symmetric expansion configuration is that for a given cross section aspect ratio the first bifurcation mode induces a top bottom asymmetry as in the 2d case whereas the subsequent ones modulate the former in the span wise direction with increasing wave numbers the asymptotic numerical method is a very efficient continuation technique judicious usage of power series analysis increases computational efficiency an hpc implementation of the resulting algorithm is presented in this paper its numerical efficiency enables to investigate problems out of reach until now span wise symmetry breaking bifurcations in a symmetric sudden expansion channel

complete surfaces with non positive extrinsic curvature in h3 and s3 abstract in this paper we give efimov and milnor s type results on complete surfaces in non euclidean space forms which satisfy that outside a compact set they have non positive gauss curvature and the square of a principal curvature function is bounded from below by a positive constant

the cauchy problem for improper affine spheres and the hessian one equation we give a conformal representation for improper affine spheres which is used to solve the cauchy problem for the hessian one equation with this representation we characterize the geodesics of an improper affine sphere study its symmetries and classify the helicoidal ones finally we obtain the complete classification of the isolated singularities of the hessian one monge ampere equation

minimal surfaces from circle patterns geometry from combinatorics the theory of polyhedral surfaces and more generally the field of discrete differential geometry are presently emerging on the border of differential and discrete geometry whereas classical differential geometry investigates smooth geometric shapes such as surfaces and discrete geometry studies geometric shapes with a finite number of elements polyhedra the theory of polyhedral surfaces aims at a development of discrete equivalents of the geometric notions and methods of surface theory the latter appears then as a limit of the refinement of the discretization current progress in this field is to a large extent stimulated by its relevance for computer graphics and visualization one of the central problems of discrete differential geometry is to find proper discrete analogues of special classes of surfaces such as minimal con stant mean curvature isothermic surfaces etc usually one can suggest vari ous discretizations with the same continuous limit which have quite different geometric properties the goal of discrete differential geometry is to find a dis cretization which inherits as many essential properties of the smooth geometry as possible our discretizations are based on quadrilateral meshes i e we discretize parametrized surfaces for the discretization of a special class of surfaces it is natural to choose an adapted parametrization in this paper we investigate conformai discretizations of surfaces i e discretizations in terms of circles and spheres and introduce a new discrete model for minimal surfaces see figures 1 and 2 in comparison with direct methods see in particular 23 leading

controllability of neutral functional integrodifferential systems in banach spaces abstract sufficient conditions for controllability of neutral functional integrodifferential systems in a banach space are established the results are obtained by using the schaefer fixed point theorem an example is provided to illustrate the theory

an extension to a theorem of jorgens calabi and pogorelov 1 1 det d2u 1 in rn must be a quadratic polynomial for n 2 a classical solution is either convex or concave the result holds without the convexity hypothesis a simpler and more analytical proof along the lines of affine geometry of the theorem was later given by cheng and yau 9 the first author extended the result for classical solutions to viscosity solutions 4 it was proven by trudinger and wang in 19 that the only open convex subset of rn which admits a convex c2 solution of det d2u 1 in with limx u x is rn in this paper we give the following extension to the theorem of jorgens calabi and pogorelov let u be a convex viscosity solution of det d2u 1 outside a bounded subset of

the affine cauchy problem abstract the aim of this paper is to solve the cauchy problem for locally strongly convex surfaces which are extremal for the equiaffine area functional these surfaces are called affine maximal surfaces and here we give a new complex representation which let us describe the solution to the corresponding cauchy problem as applications we obtain a generalized symmetry principle characterize when a curve in r 3 can be a geodesic or pre geodesic of a such surface and study the helicoidal affine maximal surfaces finally we investigate the existence and uniqueness of affine maximal surfaces with a given analytic curve in its singular set

a parallel computer implementation of the asymptotic numerical method to study thermal convection instabilities we have developed a numerical model to efficiently compute steady state combined buoyancy and thermocapillary convection solutions it features a parallel computer implementation of an asymptotic numerical method to perform steady state path following and locate bifurcation points in problems involving large size algebraic systems up to few million degrees of freedom the model has been first validated on a problem for which a reference solution exists and then is used to analyse the influence of the container size and shape on rayleigh benard marangoni convection and its related cellular pattern

ribaucour transformations for flat surfaces in the hyperbolic 3 space abstract we consider ribaucour transformations for flat surfaces in the hyperbolic 3 space h 3 we show that such transformations produce complete embedded ends of horosphere type and curves of singularities which generically are cuspidal edges moreover we prove that these ends and curves of singularities do not intersect we apply ribaucour transformations to rotational flat surfaces in h 3 providing new families of explicitly given flat surfaces h 3 which are determined by several parameters for special choices of the parameters we get surfaces that are periodic in one variable and surfaces with any even number or an infinite number of embedded ends of horosphere type

the cauchy problem for the liouville equation and bryant surfaces abstract we give a construction that connects the cauchy problem for the 2 dimensional elliptic liouville equation with a certain initial value problem for mean curvature one surfaces in hyperbolic 3 space h 3 and solve both of them we construct the unique mean curvature one surface in h 3 that passes through a given curve with a given unit normal along it and provide diverse applications in particular topics such as period problems symmetries finite total curvature planar geodesics rigidity etc are treated for these surfaces

symmetry and uniqueness of parabolic affine spheres the proof of a variety of problems about symmetry properties in partial differential equations and differential geometry is based on the maximum principle for elliptic linear equations see a1 gnn and s in this paper we apply a similar technique to a locally strongly convex parabolic affine sphere in short pa sphere embedded in the unimodular affine real 3 space j 3 the study of pa spheres is locally equivalent see c1 c2 to the study of convex solutions of the monge amp re equation

the geometric cauchy problem for the hyperbolic hessian one equation abstract we solve the problem of finding all indefinite improper affine spheres passing through a given regular curve of r 3 with a prescribed affine co normal vector field along this curve we prove the problem is well posed when the initial data are non characteristic and show that uniqueness of the solution can fail at characteristic directions as application we classify the indefinite improper affine spheres admitting a geodesic planar curve

high order predictor corrector algorithms new predictor corrector algorithms are presented for the computation of solution paths of non linear partial differential equations the predictors and the correctors are based on perturbation techniques and pade approximants this extends the asymptotic numerical method anm which is an efficient high order continuation technique without corrector the efficiency and the reliability of the new technique are assessed by several examples within thin shell theory and navier stokes equations many variants have been tested to establish an optimal algorithm

surfaces of constant curvature in r3 with isolated singularities abstract we prove that finite area isolated singularities of surfaces with constant positive curvature k 0 in r 3 are removable singularities branch points or immersed conical singularities we describe the space of immersed conical singularities of such surfaces in terms of the class of real analytic closed locally strictly convex curves in s 2 with admissible cusp singularities characterizing when the singularity is actually embedded in the global setting we describe the space of peaked spheres in r 3 i e compact convex surfaces of constant curvature k 0 with a finite number of singularities and give applications to harmonic maps and constant mean curvature surfaces

eigensensitivity analysis of rotating clamped uniform beams with the asymptotic numerical method abstract in this paper free vibrations of a rotating clamped euler bernoulli beams with uniform cross section are studied using continuation method namely asymptotic numerical method the governing equations of motion are derived using lagrange s method the kinetic and strain energy expression are derived from rayleigh ritz method using a set of hybrid variables and based on a linear deflection assumption the derived equations are transformed in two eigenvalue problems where the first is a linear gyroscopic eigenvalue problem and presents the coupled lagging and stretch motions through gyroscopic terms while the second is standard eigenvalue problem and corresponds to the flapping motion those two eigenvalue problems are transformed into two functionals treated by continuation method the asymptotic numerical method new method proposed for the solution of the linear gyroscopic system based on an augmented system which transforms the original problem to a standard form with real symmetric matrices by using some techniques to resolve these singular problems by the continuation method evolution curves of the natural frequencies against dimensionless angular velocity are determined at high angular velocity some singular points due to the linear elastic assumption are computed numerical tests of convergence are conducted and the obtained results are compared to the exact values results obtained by continuation are compared to those computed with discrete eigenvalue problem

complete affine hypersurfaces part i the completeness of affine metrics on etudie les proprietes des hypersurfaces invariantes sous le groupe des transformations affines unimodulaires

singularities of improper affine maps and their hessian equation abstract we study improper affine spheres with some admissible singularities called improper affine maps and associated to the unimodular hessian equation in particular we characterize when a curve of r 3 is the singular curve of some improper affine map with prescribed cuspidal edges and swallowtails also we consider improper affine maps with isolated singularities and show some similarities and differences between the hessian 1 equation and the hessian 1 equation as a consequence we construct global examples with the desired singularities

asymptotic numerical methods and pade approximants for non linear elastic structures in this paper we apply asymptotic numerical methods for computing non linear equilibrium paths of elastic beam plate and shell structures the non linear branches are sought in the form of asymptotic expansions and they are determined by solving numerically fem several linear problems with a single stiffness matrix a large number of terms of the series can be easily computed by using recurrence formulas in comparison with a more classical step by step procedure the method is rapid and automatic we show with some examples that the choice of the expansion s parameter and the use of pade approximants play an important role in the determination of the size of the domain of convergence

power series analysis as a major breakthrough to improve the efficiency of asymptotic numerical method in the vicinity of bifurcations this paper presents the outcome of power series analysis in the framework of the asymptotic numerical method we theoretically demonstrate and numerically evidence that the emergence of geometric power series in the vicinity of simple bifurcation points is a generic behavior so we propose to use this hallmark as a bifurcation indicator to locate and compute very efficiently any simple bifurcation point finally a power series that recovers an optimal step length is build in the neighborhood of bifurcation points the reliability and robustness of this powerful approach is then demonstrated on two application examples from structural mechanics and hydrodynamics

complete linear weingarten surfaces of bryant type a plateau problem at infinity in this paper we study a large class of weingarten surfaces which includes the constant mean curvature one surfaces and flat surfaces in the hyperbolic 3 space we show that these surfaces can be parametrized by holomorphic data like minimal surfaces in the euclidean 3 space and we use it to study their completeness we also establish some existence and uniqueness theorems by studing the plateau problem at infinity when is a given curve on the ideal boundary the asymptotic boundary of a complete surface in our family and how many embedded solutions are there

bjorling problem for timelike surfaces in the lorentz minkowski space abstract we solve the bjorling problem for timelike surfaces in the lorentz minkowski space through a split complex representation formula obtained for this kind of surface our approach uses the split complex numbers and natural split holomorphic extensions as applications we show that the minimal timelike surfaces of revolution as well as minimal ruled timelike surfaces can be characterized as solutions of certain adequate bjorling problems in the lorentz minkowski space

entire solutions of the degenerate monge ampere equation with a finite number of singularities we determine the global behavior of every c2 solution to the two dimensional degenerate monge ampere equation uxxuyy uxy2 0 over the finitely punctured plane with this we classify every solution in the once or twice punctured plane moreover when we have more than two singularities if the solution u is not linear in a half strip we obtain that the singularities are placed at the vertices of a convex polyhedron p and the graph of u is made by pieces of cones outside of p which are suitably glued along the sides of the polyhedron finally if we look for analytic solutions then there is at most one singularity and the graph of u is either a cylinder no singularity or a cone one singularity

the geometry of fronts we shall introduce the singular curvature function on cuspidal edges of surfaces which is related to the gauss bonnet formula and which characterizes the shape of cuspidal edges moreover it is closely related to the behavior of the gaussian curvature of a surface near cuspidal edges and swallowtails