forced vibrations of superquadratic hamiltonian systems on etudie l existence des solutions t periodiques ter t 0 du systeme hamiltonien suivant z gh z f t ou g est la matrice asymetrique standard

free vibrations for a semilinear wave equation abstract existence and regularity of nontrivial time periodic solutions are proved for semilinear wave equations under mild smoothness monotonicity and superlinearity assumptions the forced vibration case is also treated

dirichlet problem for semilinear hyperbolic equation multidimensional case abstract we discuss the solvability of the dirichlet problem for the semilinear equation of the vibrating string x t t t y x t y f t y x t y 0 in higher dimensions with sides length being irrational numbers and superlinear nonlinearity to this effect we derive a new dual variational method

a new proof and an extension of a theorem of p rabinowitz concerning nonlinear wave equations abstract in 1978 p rabinowitz proved a theorem concerned with the existence of a nontrivial periodic solution of a nonlinear wave equation with a continuous increasing superlinear nonlinear term in this paper we present a new and simpler proof of this theorem and relax an assumption on the nonlinear term which is a discontinuous nondecreasing function author

perturbation theory for anisotropic dielectric interfaces and application to subpixel smoothing of discretized numerical methods we derive a correct first order perturbation theory in electromagnetism for cases where an interface between two anisotropic dielectric materials is slightly shifted most previous perturbative methods give incorrect results for this case even to lowest order because of the complicated discontinuous boundary conditions on the electric field at such an interface our final expression is simply a surface integral over the material interface of the continuous field components from the unperturbed structure the derivation is based on a localized coordinate transformation technique which avoids both the problem of field discontinuities and the challenge of constructing an explicit coordinate transformation by taking the limit in which the coordinate perturbation is infinitesimally localized around the boundary not only is our result potentially useful in evaluating boundary perturbations e g from fabrication imperfections in highly anisotropic media such as many metamaterials but it also has a direct application in numerical electromagnetism in particular we show how it leads to a subpixel smoothing scheme to ameliorate staircasing effects in discretized simulations of anisotropic media in such a way as to greatly reduce the numerical errors compared to other proposed smoothing schemes

abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity critical point theorems and applications to some nonlinear problems with strong resonance at infinity p bartolo and v benci istituto di matematica applicata via re david x0 bari ital and istituto di analisi matematica palazzo ateneo l ia scolai 2 bari italy received in revised form 20 december 1952

normalized cuts and image segmentation we propose a novel approach for solving the perceptual grouping problem in vision rather than focusing on local features and their consistencies in the image data our approach aims at extracting the global impression of an image we treat image segmentation as a graph partitioning problem and propose a novel global criterion the normalized cut for segmenting the graph the normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups we show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion we applied this approach to segmenting static images as well as motion sequences and found the results to be very encouraging

a stable fdtd algorithm for non diagonal anisotropic dielectrics a stable fdtd algorithm is developed for simulating maxwell s equations in anisotropic dielectric materials with principal axes not aligned with the grid the algorithm is stable because the finite difference operator that converts d to e is symmetric and positive semidefinite for contrast a previously developed asymmetric algorithm is shown to suffer from late time instabilities the presented algorithm has second order error for continuous dielectric materials and the error can be reduced to third order by richardson extrapolation applied to dielectrics with sharp interfaces the algorithm has first order error even when averaging the dielectric within partially filled grid cells however averaging the dielectric permits richardson extrapolation to obtain second order error

approximate pairwise clustering for large data sets via sampling plus extension pairwise clustering methods have shown great promise for many real world applications however the computational demands of these methods make them impractical for use with large data sets the contribution of this paper is a simple but efficient method called espec that makes clustering feasible for problems involving large data sets our solution adopts a sampling clustering plus extension strategy the methodology starts by selecting a small number of representative samples from the relational pairwise data using a selective sampling scheme then the chosen samples are grouped using a pairwise clustering algorithm combined with local scaling and finally the label assignments of the remaining instances in the data are extended as a classification problem in a low dimensional space which is explicitly learned from the labeled samples using a cluster preserving graph embedding technique extensive experimental results on several synthetic and real world data sets demonstrate both the feasibility of approximately clustering large data sets and acceleration of clustering in loadable data sets of our method

nonlinear problems with strong resonance at infinity an abstract theorem and applications on considere l equation au n u u e ou a est un operateur lineaire n avec c 1 e r et e est un espace de hilbert on suppose que n a une derivee a l infini n et que o appartient au spectre de a n on demontre un theoreme pour la multiplicite des solutions

periodic and quasi periodic solutions of nonlinear wave equations via kam theory in this paper the nonlinear wave equation u u u xx v x u x t varepsilon u 3 x t 0 is studied it is shown that for a large class of potentials v x one can use kam methods to construct periodic and quasi periodic solutions in time for this equation

existence of infinitely many periodic solutions for the radially symmetric wave equation with resonance abstract in this paper we consider the periodic dirichlet problem for a forced nonlinear wave equation with resonance u t t u u a t x u p 1 u in a n dimensional ball under some suitable assumptions on p and a t x we prove the existence of infinitely many radially symmetric time periodic solutions for the problem by variational methods

a new kind of integral transform and its application to periodic solutions in this paper we suggest a new kind of integral transform and establish the main theorems of that this approach allows us to show some general existence theorems for periodic solutions to the semilinear wave equation with the forced term having double frequency a novelty of this transform is to prove the existence of a combination oscillation for the continuous system without the so called difficulty of small divisors by j j stoker nonlinear vibrations interscience new york 1950 in addition our integral transform can be applied to rather general reaction diffusion systems however as a concrete example we show that maginu s model f rothe j math biol 7 1979 375 384 has multiple nonconstant t periodic plane wave solutions

topological results on a certain class of functionals and application abstract we study in this paper problems of the type u u p 1 u x bounded r n u 0 i where x is given and where p 1 n 2 n 2 p 1 if n 2 our main result is that i has an infinite number of solutions for a residual set of in h 1 in particular for many n n there exists an open and dense subset of in h 1 such that i has n distinct solutions for such an this result is to be related to the conjecture developed in 1 of the existence of an infinite number of solutions to i the proof relies on a general characterization of level sets for a certain class of functionals when there are no critical value in a large enough interval in addition to the study of problem i we apply this characterization to give another proof using e g brouwer s fixed point theorem for some classical results about even functionals and saddle points

periodic solutions of symmetric wave equations this paper concerns the existence of solutions to the following nonautonomous wave equation gu uttu f 2 f x t is nondecreasing in and f x t 0 if and only if 0 our first result deals with the case where f is superlinear at theorem 1 let f 1 and f 2 be satisfied assume moreover that f satisfies f 3 there are r 0 and 2 such that

a survey of kernel and spectral methods for clustering clustering algorithms are a useful tool to explore data structures and have been employed in many disciplines the focus of this paper is the partitioning clustering problem with a special interest in two recent approaches kernel and spectral methods the aim of this paper is to present a survey of kernel and spectral clustering methods two approaches able to produce nonlinear separating hypersurfaces between clusters the presented kernel clustering methods are the kernel version of many classical clustering algorithms e g k means som and neural gas spectral clustering arise from concepts in spectral graph theory and the clustering problem is configured as a graph cut problem where an appropriate objective function has to be optimized an explicit proof of the fact that these two paradigms have the same objective is reported since it has been proven that these two seemingly different approaches have the same mathematical foundation besides fuzzy kernel clustering methods are presented as extensions of kernel k means clustering algorithm

modeling of sloped interfaces on a yee grid to represent material boundaries in the finite difference time domain or frequency domain method effective cell permittivity eeff can be introduced for each grid cell crossed by material interface in this paper we revisit the derivation of tensorial eeff for a sloped interface and describe possible interpolation schemes for coupling of different effective electric field and induction components near the interface we put the resulting non symmetric and symmetrized effective permittivity matrices to numerical tests in the frequency domain for very high contrast interfaces the symmetrized schemes perform worse than simple staircasing while non symmetrized interpolation retains the second order convergence

periodic solutions for completely resonant nonlinear wave equations with dirichlet boundary conditions we consider the nonlinear string equation with dirichlet boundary conditions utt uxx u with u u3 o u5 odd and analytic 0 and we construct small amplitude periodic solutions with frequency for a large lebesgue measure set of close to 1 this extends previous results where only a zero measure set of frequencies could be treated the ones for which no small divisors appear the proof is based on combining the lyapunov schmidt decomposition which leads to two separate sets of equations dealing with the resonant and non resonant fourier components respectively the q and the p equations with resummation techniques of divergent powers series allowing us to control the small divisors problem the main difficulty with respect to the nonlinear wave equations utt uxx mu u m 0 is that not only the p equation but also the q equation is infinite dimensional

infinitely many periodic solutions for a semilinear wave equation in a ball in rn abstract in this paper we study the wave equation with sub linearity in a ball with periodic dirichlet conditions combining the minimax principle with an approximation argument we prove the existence of infinitely many time periodic solutions for the problem the proof also relies on the structure of the spectrum of the linearized problem

oscillatory phenomena associated to semilinear wave equations in one spatial dimension let g be a nonincreasing odd c function and i 0 we establish that for any solution u s c r hq 0 of the equation utt uxx g u 0 and any io 0 l the function t u t xo satisfies the following alternative either u t x0 0 vt g r or vo r there exist t and t2 in a a 21 such that u ti xo 0 and u t2 xo 0 we study the structure of the set of points satisf ing the first possibility we give analogous results for ux and for some other homogeneous boundary conditions

numerical stability of nonorthogonal fdtd methods in this paper a sufficient test for the numerical stability of generalized grid finite difference time domain fdtd schemes is presented it is shown that the projection operators of such schemes must be symmetric positive definite without this property such schemes can exhibit late time instabilities the origin and the characteristics of these late time instabilities are also uncovered based on this study nonorthogonal grid fdtd schemes nfdtd and the generalized yee gy methods are proposed that are numerically stable in the late time for quadrilateral prism elements allowing these methods to be extended to problems requiring very long time simulations the study of numerical stability that is presented is very general and can be applied to most solutions of maxwell s equations based on explicit time domain schemes

periodic solutions of nonlinear wave equations on obtient des conditions suffisantes pour l existence de solutions doublement periodiques de l equation d onde non lineaire u ts f t s u u t t s u t s u t s t ou f est une fonction continue donnee dans r 3 et f est t periodique en t et s

a new index theory for linear self adjoint operator equations and its applications abstract we develop an index theory for linear self adjoint operator equation with the linear self adjoint operator having no compact resolvent as applications we consider the existence and multiplicity of periodic solutions of wave equations and beam equations

cantor families of periodic solutions for wave equations via a variational principle abstract we prove existence of small amplitude periodic solutions of completely resonant wave equations with frequencies in a cantor set of asymptotically full measure via a variational principle a lyapunov schmidt decomposition reduces the problem to a finite dimensional bifurcation equation variational in nature defined on a cantor set of non resonant parameters the cantor gaps are due to small divisors phenomena to solve the bifurcation equation we develop a suitable variational method in particular we do not require the typical arnold non degeneracy condition of the known theory on the nonlinear terms as a consequence our existence results hold for new generic sets of nonlinearities

a 3 d tensor fdtd formulation for treatment of sloped interfaces in electrically inhomogeneous media a general integral finite difference time domain fdtd formulation on cubical grids for the modeling of electrically inhomogeneous media of arbitrary shape is derived the simple cubical structure is maintained and no modifications of the integral paths are necessary instead a three dimensional tensor relationship between the average electric flux density and electric field is determined beforehand and used during the simulation to account for discontinuities in the neighborhood of sloped interfaces the accuracy of the method is confirmed by comparisons with analytical solutions

a property of exponential stability in nonlinear wave equations near the fundamental linear mode abstract we consider the nonlinear wave equation utt c2uxx u u t 0 u t 0 where is an analytic function satisfying 0 0 and 00 0 for each value of the harmonic energy e i e energy when 0 we consider the closed curve e which is the phase space trajectory of the linear mode with lowest frequency if the perturbation parameter and the harmonic energy e are small enough then near e we construct by means of a suitable simplified equation a closed curve e with the following property provided the distance between the initial datum and such a curve is small enough the corresponding solution remains close to e up to times exponentially long with 1 eve cannot expect the curves e to be trajectories of solutions of such curves depend smoothly on the parameters e and the above result is deduced from a general theorem on generic hamiltonian perturbations of completely resonant linear systems

optimal cluster preserving embedding of nonmetric proximity data for several major applications of data analysis objects are often not represented as feature vectors in a vector space but rather by a matrix gathering pairwise proximities such pairwise data often violates metricity and therefore cannot be naturally embedded in a vector space concerning the problem of unsupervised structure detection or clustering in this paper a new embedding method for pairwise data into euclidean vector spaces is introduced we show that all clustering methods which are invariant under additive shifts of the pairwise proximities can be reformulated as grouping problems in euclidian spaces the most prominent property of this constant shift embedding framework is the complete preservation of the cluster structure in the embedding space restating pairwise clustering problems in vector spaces has several important consequences such as the statistical description of the clusters by way of cluster prototypes the generic extension of the grouping procedure to a discriminative prediction rule and the applicability of standard preprocessing methods like denoising or dimensionality reduction

free and forced vibrations of nonlinear wave equations in ball first we shall deal with the free vibrations of a nonlinear radially symmetric wave equation t2 u f r u in n dimensional ball ba with center at the origin and radius a where f is smooth monotone decreasing in u and satisfies f r 0 0 f r u has asymptotic properties f r u o u3 u 0andu for n 1 3 we shall show the existence of infinitely many radially symmetric time periodic solutions with different periods of irrational multiple of a second we shall deal with bvp for a forced nonlinear wave equation t2 u eg r t u where g is t periodic in t and e is a small parameter under some diophantine condition on a t we shall show the existence of time periodic solutions of the bvp for 1 n 5 we shall construct infinitely many t satisfying the above diophantine inequality using asymptotic expansions of the zero points of the bessel functions

vector quantization based approximate spectral clustering of large datasets spectral partitioning recently popular for unsupervised clustering is infeasible for large datasets due to its computational complexity and memory requirement therefore approximate spectral clustering of data representatives selected by various sampling methods was used alternatively we propose to use neural networks self organizing maps and neural gas which are shown successful in quantization with small distortion as preliminary sampling for approximate spectral clustering asc we show that they usually outperform k means sampling which was shown superior to various sampling methods in terms of clustering accuracy obtained by asc more importantly for quantization based asc we introduce a local density based similarity measure constructed without any user set parameter which achieves accuracies superior to the accuracies of commonly used distance based similarity

a three dimensional fdtd algorithm in curvilinear coordinates em scattering the finite difference time domain fdtd algorithm for the solution of electromagnetic scattering problems is formulated in numerically defined generalized coordinates in three dimensions and implemented in a code with the lowest order bayliss turkel radiation boundary condition expressed in spherical coordinates it is shown that the algorithm is capable of accurately tracking the progress of a pulse of electromagnetic radiation through the curvilinear mesh generated by a body of revolution the only problems occurring in the vicinity of the rotation axis which represents a coordinate singularity a simple method to deal with this singular line is presented and discussed and its is shown that at least for the test problem this approximation is sufficient the algorithm discussed is useful for the solution of the exterior problem in the presence of conductors and dielectrics with complicated shapes and electrical compositions and for near field problems such as cavity penetration problems the far fields are obtained by replacing the scatterer with a virtual surface enclosing all sources

a unified framework for representation based subspace clustering of out of sample and large scale data under the framework of spectral clustering the key of subspace clustering is building a similarity graph which describes the neighborhood relations among data points some recent works build the graph using sparse low rank and ell 2 norm based representation and have achieved the state of the art performance however these methods have suffered from the following two limitations first the time complexities of these methods are at least proportional to the cube of the data size which make those methods inefficient for solving the large scale problems second they cannot cope with the out of sample data that are not used to construct the similarity graph to cluster each out of sample datum the methods have to recalculate the similarity graph and the cluster membership of the whole data set in this paper we propose a unified framework that makes the representation based subspace clustering algorithms feasible to cluster both the out of sample and the large scale data under our framework the large scale problem is tackled by converting it as the out of sample problem in the manner of sampling clustering coding and classifying furthermore we give an estimation for the error bounds by treating each subspace as a point in a hyperspace extensive experimental results on various benchmark data sets show that our methods outperform several recently proposed scalable methods in clustering a large scale data set

diophantine approximation bessel functions and radially symmetric periodic solutions of semilinear wave equations in a ball the aim of this paper is to consider the radially symmetric periodic dirichlet problem on 0 t x b n a for the equation u tt deltau f t x u where delta is the classical laplacian operator and b n a denotes the open ball of center 0 and radius a in r n when alpha a t is a sufficiently large irrational with bounded partial quotients we combine some number theory techniques with the asymptotic properties of the bessel functions to show that 0 is not an accumulation point of the spectrum of the linear part this result is used to obtain existence conditions for the nonlinear problem

computations of critical groups and applications to asymptotically linear wave equation and beam equation in this paper by using the morse index theory for strongly indefinite functionals developed in nonlinear anal tma in press we compute precisely the critical groups at the origin and at infinity respectively the abstract theorems are used to study the existence multiplicity of nontrivial periodical solutions for asymptotically wave equation and beam equation with resonance both at infinity and at zero

standing waves on an infinitely deep perfect fluid under gravity the existence of two dimensional standing waves on the surface of an infinitely deep perfect fluid under gravity is established when formulated as a second order equation for a real valued function w on the 2 torus and a positive parameter the problem is fully nonlinear the highest order x derivative appears in the nonlinear term but not in the linearization at 0 and completely resonant there are infinitely many linearly independent eigenmodes of the linearization at 0 for all rational values of the parameter moreover for any prescribed order of accuracy there exists an explicit approximate solution of the nonlinear problem in the form of a trigonometric polynomial using a nash moser method to seek solutions of the nonlinear problem as perturbations of the approximate solutions the existence question can be reduced to one of estimating the inverses of linearized operators at non zero points after changing coordinates these operators become first order and non local in space and second order in time after further changes of variables the main parts become diagonal with constant coefficients and the remainder is regularizing or quasi one dimensional in the sense of 22 the operator can then be inverted for two reasons first the explicit formula for the approximate solution means that restricted to the infinite dimensional kernel of the linearization at zero the inverse exists and can be estimated second the small divisor problems that arise on the complement of this kernel can be overcome by considering only particular parameter values selected according to their diophantine properties a parameter dependent version of the nash moser implicit function theorem now yields the existence of a set of unimodal standing waves on flows of infinite depth corresponding to a set of values of the parameter 1 which is dense at 1 unimodal means that the term of smallest order in the amplitude is cos x cos t which is one of many eigenfunctions of the completely resonant linearized problem

periodic solutions to nonlinear one dimensional wave equation with dependent coefficients this paper deals with t periodicity and regularity of solutions to the one dimensional nonlinear wave equation with x dependent coefficients

approximate spectral clustering with utilized similarity information using geodesic based hybrid distance measures spectral clustering has been popular thanks to its ability to extract clusters of varying characteristics without using a parametric model in expense of high computational cost required for eigendecomposition of pairwise similarities in order to utilize its advantages in large datasets where it is infeasible due to its computational burden approximate spectral clustering asc methods apply spectral clustering on a reduced set of points data representatives selected by sampling or quantization this two step approach i e finding the representatives and then clustering them brings new opportunities for precise similarity definition such as manifold based topological relations data distribution within the voronoi polyhedra of the representatives and their geodesic distance information which are often ignored in similarity definition for asc in this study we propose geodesic based hybrid similarity criteria which enable the use of different types of information for accurate similarity representation in asc despite the fact that geodesic concept has been widely used in clustering our contribution is the unique way of representing data topology to form geodesic relations and jointly harnessing various information types including topology distance and density the proposed criteria are tested using both sampling selective sampling and quantization neural gas and k means approaches experiments on artificial datasets well known small medium size real datasets and four large datasets four remote sensing images with different types of clusters show that the proposed geodesic based hybrid similarity criteria outperform traditional similarity criteria in terms of clustering accuracies and several cluster validity indices highlightsgeodesic based hybrid similarity measures are proposed for approximate spectral clustering neighborhood graph required for geodesic approach is effectively constructed by weighted delaunay triangulation conn the proposed geodesic based hybrid similarities outperform in terms of both accuracies and cluster validity indices the proposed geodesic based hybrid similarities can be powerful for clustering of large remote sensing images the proposed similarities are significant especially for quantization based approximate spectral clustering

learning and matching of dynamic shape manifolds for human action recognition in this paper we learn explicit representations for dynamic shape manifolds of moving humans for the task of action recognition we exploit locality preserving projections lpp for dimensionality reduction leading to a low dimensional embedding of human movements given a sequence of moving silhouettes associated to an action video by lpp we project them into a low dimensional space to characterize the spatiotemporal property of the action as well as to preserve much of the geometric structure to match the embedded action trajectories the median hausdorff distance or normalized spatiotemporal correlation is used for similarity measures action classification is then achieved in a nearest neighbor framework to evaluate the proposed method extensive experiments have been carried out on a recent dataset including ten actions performed by nine different subjects the experimental results show that the proposed method is able to not only recognize human actions effectively but also considerably tolerate some challenging conditions e g partial occlusion low quality videos changes in viewpoints scales and clothes within class variations caused by different subjects with different physical build styles of motion etc

fast monte carlo algorithms for matrices ii computing a low rank approximation to a matrix in many applications the data consist of or may be naturally formulated as an m times n matrix a it is often of interest to find a low rank approximation to a i e an approximation d to the matrix a of rank not greater than a specified rank k where k is much smaller than m and n methods such as the singular value decomposition svd may be used to find an approximation to a which is the best in a well defined sense these methods require memory and time which are superlinear in m and n for many applications in which the data sets are very large this is prohibitive two simple and intuitive algorithms are presented which when given an m times n matrix a compute a description of a low rank approximation d to a and which are qualitatively faster than the svd both algorithms have provable bounds for the error matrix a d for any matrix x let x f and x 2 denote its frobenius norm and its spectral norm respectively in the first algorithm c columns of a are randomly chosen if the m times c matrix c consists of those c columns of a after appropriate rescaling then it is shown that from c tc approximations to the top singular values and corresponding singular vectors may be computed from the computed singular vectors a description d of the matrix a may be computed such that mathrm rank d le k and such that left a d right xi 2 le min d mathrm rank d le k left a d right xi 2 poly k 1 c left a right 2 f holds with high probability for both xi 2 f this algorithm may be implemented without storing the matrix a in random access memory ram provided it can make two passes over the matrix stored in external memory and use o cm c 2 additional ram the second algorithm is similar except that it further approximates the matrix c by randomly sampling r rows of c to form a r times c matrix w thus it has additional error but it can be implemented in three passes over the matrix using only constant additional ram to achieve an additional error beyond the best rank k approximation that is at most epsilon a 2 f both algorithms take time which is polynomial in k 1 epsilon and log 1 delta where delta 0 is a failure probability the first takes time linear in mbox max m n and the second takes time independent of m and n our bounds improve previously published results with respect to the rank parameter k for both the frobenius and spectral norms in addition the proofs for the error bounds use a novel method that makes important use of matrix perturbation theory the probability distribution over columns of a and the rescaling are crucial features of the algorithms which must be chosen judiciously

on strongly indefinite functionals with applications recently in their remarkable paper critical point theory for indefinite functionals v benci and p rabinowitz gave a direct approach avoiding finitedimensional approximationsto the existence theory for critical points of indefinite functionals in this paper we develop under weaker hypotheses a simpler but more general theory for such problems in the second part of the paper the abstract results are applied to a class of resonance problems of the landesman and lazer type and moreover they are illustrated by an application to a wave equation problem 0 introduction this paper is devoted to the critical point theory of strongly indefinite functionals in a real hilbert space h here a strongly indefinite functional is a cl map h r which is neither bounded from above or below not even modulo a finite dimensional subspace such functionals arise for example in the study of periodic solutions of the one dimensional wave equation in the study of periodic solutions of hamiltonian systems of ordinary differential equations in the existence theory for systems of elliptic equations etc to find the critical points of strongly indefinite functionals different methods were used for example rabinowitz studied the restrictions of the functional to finite dimensional subspaces 1 2 and then he tried to find the critical points of the original problem by passing to the limit for a suitable subsequence of the critical points of the restricted problems amann 3 reduced the indefinite functionals by means of a saddlepoint reduction to a finite dimensional problem of course this is only possible under some strong hypotheses under certain assumptions the functional can be replaced by a dual variational problem having some better properties see for example 4 or 5 the first direct approach was given by benci and rabinowitz 6 the essential in their paper is the observation that more detailed information about the special form of the deformations a substitute for the gradient flow associated to the functional is needed they constructed deformations having special representations by solving appropriate differential equations approximately by time discretisation this method has some disadvantages because one needs strong hypotheses in order to get the approximations uniformly on bounded sets a main part of our paper will be the construction of flows and deformations having special representations by solving appropriate differential equations exactly received by the editors march 30 1981 and in revised form november 30 1981 1980 mathematics subject classification primary 58e05 secondary 47h25 35l05