nonlinear scattering theory at low energy abstract we study the scattering theory of a conservative nonlinear one parameter group of operators on a hilbert space x relative to a group of linear unitary operators under certain hypotheses the scattering operator carries a neighborhood of 0 in x into x the theory is designed to apply to the semilinear schrodinger and klein gordon equations

global existence for the einstein vacuum equations in wave coordinates we prove global stability of minkowski space for the einstein vacuum equations in harmonic wave coordinate gauge for the set of restricted data coinciding with the schwarzschild solution in the neighborhood of space like infinity the result contradicts previous beliefs that wave coordinates are unstable in the large and provides an alternative approach to the stability problem originally solved for unrestricted data in a different gauge and with a precise description of the asymptotic behavior at null infinity by d christodoulou and s klainerman

global existence of classical solutions to the minimal surface equation in two space dimensions with slow decay initial value in this paper we prove the global existence of classical solutions to the cauchy problem for the minimal surface equation in the minkowski space r1 2 with slow decay initial value

kernel density estimation with spherical data summary we study two natural classes of kernel density estimators for use with spherical data members of both classes have already been used in practice the classes have an element in common but for the most part they are disjoint however all members of the first class are asymptotically equivalent to one another and to a single element of the second class in this sense the second class contains the first it includes some estimators which out perform all those in the first class if loss is measured in either squared error or kullback leibler senses explicit formulae are given for bias variance and loss and large sample properties of these quantities are described numerical illustrations are presented

remarks on the global sobolev inequalities in the minkowski space rn 1 on demontre des inegalites de sobolev globales sur l espace de minkowski r n 1 le theoreme principal est valable sur des tranches a temps fixe

global existence of null form wave equations on small asymptotically euclidean manifolds abstract we prove the global existence of the small solutions to the cauchy problem for quasilinear wave equations satisfying the null condition on r 3 g where the metric g is a small perturbation of the flat metric and approaches the euclidean metric like 1 x 2 2 with 1 global and almost global existence for systems without the null condition are also discussed for certain small time dependent perturbations of the flat metric in appendix a

global existence for systems of quasilinear wave equations in 1 4 dimensions abstract hormander proved global existence of solutions for sufficiently small initial data for scalar wave equations in 1 4 dimensions of the form u q u u u where q vanishes to second order and u 2 q 0 0 0 0 without the latter condition only almost global existence may be guaranteed the first author and sogge considered the analog exterior to a star shaped obstacle both results relied on writing the lowest order terms u u 1 2 u 2 and as such do not immediately generalize to systems the current study remedies such and extends both results to the case of multiple speed systems

uniform bound of sobolev norms of solutions to 3d nonlinear wave equations with null condition abstract this article concerns the time growth of sobolev norms of classical solutions to the 3d quasi linear wave equations with the null condition given initial data in h s h s 1 with compact supports the global well posedness theory has been established independently by klainerman 13 and christodoulou 3 respectively for a relatively large integer s however the highest order sobolev energy namely the h s energy of solutions may have a logarithmic growth in time in this paper we show that the h s energy of solutions is also uniformly bounded for s 5 the proof employs the generalized energy method of klainerman enhanced by weighted l 2 estimates and the ghost weight introduced by alinhac

regularity of solutions to nonlinear dispersive equations on considere le probleme de cauchy pour les equations dispersives non lineaires de la forme t u ip d u f u x r n t u x 0 u 0 x

the possibilistic c means algorithm insights and recommendations recently the possibilistic c means algorithm pcm was proposed to address the drawbacks associated with the constrained memberships used in algorithms such as the fuzzy c means fcm in this issue barni et al 1996 report a difficulty they faced while applying the pcm and note that it exhibits an undesirable tendency to converge to coincidental clusters the purpose of this paper is not just to address the issues raised by barni et al but to go further and analytically examines the underlying principles of the pcm and the possibilistic approach in general we analyze the data sets used by barni et al and interpret the results reported by them in the light of our findings

weighted l and l1 estimates for solutions to the classical wave equation in three space dimensions on considere le probleme aux valeurs initiales u 0 u x 0 0 u t x 0 g x on donne des estimations l et l 1 pour les solutions dans le cas general a symetrie non spherique

a contribution to convergence theory of fuzzy c means and derivatives in this paper we revisit the convergence and optimization properties of fuzzy clustering algorithms in general and the fuzzy c means fcm algorithm in particular our investigation includes probabilistic and a slightly modified implementation of possibilistic memberships which will be discussed under a unified view we give a convergence proof for the axis parallel variant of the algorithm by gustafson and kessel that can be generalized to other algorithms more easily than in the usual approach using reformulated fuzzy clustering algorithms we apply banach s classical contraction principle and establish a relationship between saddle points and attractive fixed points for the special case of fcm we derive a sufficient condition for fixed points to be attractive allowing identification of them as local minima of the objective function excluding the possibility of a saddle point

a shift in the strauss exponent for semilinear wave equations with a not effective damping abstract in this note we study the global existence of small data solutions to the cauchy problem for the semilinear wave equation with a not effective scale invariant damping term namely v t t v 2 1 t v t v p v 0 x v 0 x v t 0 x v 1 x where p 1 n 2 we prove blow up in finite time in the subcritical range p 1 p 2 n and existence theorems for p p 2 n n 2 3 in this way we find the critical exponent for small data solutions to this problem our results lead to the conjecture p 2 n p 0 n 2 for n 2 where p 0 n is the strauss exponent for the classical semilinear wave equation with power nonlinearity

on almost global existence for nonrelativistic wave equations in 3d almost global solutions are constructed to three dimensional quadratically nonlinear wave equations the proof relies on generalized energy estimates and a new decay estimate the method applies to equations that are only classically invariant such as the nonlinear system of hyperelasticity 1996 john wiley sons inc

global solutions for the gravity water waves equation in dimension 3 we show existence of global solutions for the gravity water waves equation in dimension 3 in the case of small data the proof combines energy estimates which yield control of l 2 related norms with dispersive estimates which give decay in l 8 to obtain these dispersive estimates we use an analysis in fourier space the study of space and time resonances is then the crucial point

applying the possibilistic c means algorithm in kernel induced spaces in this paper we study a kernel extension of the classic possibilistic c means in the proposed extension we implicitly map input patterns into a possibly high dimensional space by means of positive semidefinite kernels in this new space we model the mapped data by means of the possibilistic clustering algorithm we study in more detail the special case where we model the mapped data using a single cluster only since it turns out to have many interesting properties the modeled memberships in kernel induced spaces yield a modeling of generic shapes in the input space we analyze in detail the connections to one class support vector machines and kernel density estimation thus suggesting that the proposed algorithm can be used in many scenarios of unsupervised learning in the experimental part we analyze the stability and the accuracy of the proposed algorithm on some synthetic and real datasets the results show high stability and good performances in terms of accuracy

stability based validation of clustering solutions data clustering describes a set of frequently employed techniques in exploratory data analysis to extract natural group structure in data such groupings need to be validated to separate the signal in the data from spurious structure in this context finding an appropriate number of clusters is a particularly important model selection question we introduce a measure of cluster stability to assess the validity of a cluster model this stability measure quantifies the reproducibility of clustering solutions on a second sample and it can be interpreted as a classification risk with regard to class labels produced by a clustering algorithm the preferred number of clusters is determined by minimizing this classification risk as a function of the number of clusters convincing results are achieved on simulated as well as gene expression data sets comparisons to other methods demonstrate the competitive performance of our method and its suitability as a general validation tool for clustering solutions in real world problems

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

support vector clustering we present a novel clustering method using the approach of support vector machines data points are mapped by means of a gaussian kernel to a high dimensional feature space where we search for the minimal enclosing sphere this sphere when mapped back to data space can separate into several components each enclosing a separate cluster of points we present a simple algorithm for identifying these clusters the width of the gaussian kernel controls the scale at which the data is probed while the soft margin constant helps coping with outliers and overlapping clusters the structure of a dataset is explored by varying the two parameters maintaining a minimal number of support vectors to assure smooth cluster boundaries we demonstrate the performance of our algorithm on several datasets

the lifespan for nonlinear wave equations with multiple propagation speeds in 3d abstract in this paper we consider the cauchy problem for systems of nonlinear wave equations with multiple propagation speeds in three space dimensions on the basis of some klainerman sideris type weighted estimates we get the lifespan of small amplitude solutions for systems with nonlinearities depending on both the unknown functions and their derivatives

almost global existence to nonlinear wave equations in three space dimensions on considere l existence a t grand de solutions de petite amplitude pour des equations d onde d ordre 2 non lineaires a 4 dimensions d espace temps

nonresonance and global existence of prestressed nonlinear elastic waves this article considers the existence of global classical solutions to the cauchy problem in nonlinear elastodynamics the unbounded elastic medium is assumed to be homogeneous isotropic and hyperelastic as in the theory of 3d nonlinear wave equations in three space dimensions global existence hinges on two basic assumptions first the initial deformation must be a small displacement from equilibrium in this case a prestressed homogeneous dilation of the reference conflguration and equally important the nonlinear terms must obey a type of nonresonance or null condition the omission of either of these assumptions can lead to the breakdown of solutions in flnite time in particular nonresonance complements the genuine nonlinearity condition of f john under which arbitrarily small spherically symmetric displacements develop singularities although one expects this to carry over to the nonsymmetric case as well 4 john also showed that small solutions exist almost globally 5 see also 10 formation of singularities for large displacements was illustrated by tahvildar zadeh 16 the nonresonance condition introduced here represents a substantial improvement over our previous work on this topic 13 to explain the difierence roughly our earlier version of the null condition forced the cancellation of all nonlinear wave interactions to flrst order along the characteristic cones here only the cancellation of nonlinear wave interactions among individual wave families is required the di culty in realizing this weaker version is that the decomposition of elastic waves into their longitudinal and transverse components involves the nonlocal helmholtz projection which is ill suited to nonlinear analysis however our decay estimates make clear that only the leading contribution of the resonant interactions along the characteristic cones is potentially dangerous and this permits the usage of approximate local decompositions

on one dimension semi linear wave equations with null conditions abstract it is well known that in space dimensions at least three semilinear wave equations with null conditions admit global solutions for small initial data it is also known that in space dimension two such result still holds for a certain class of quasi linear wave equations with null conditions the proofs are based on the decay mechanism of linear waves however in one space dimension waves do not decay nevertheless we will prove that small data still lead to global solutions if the null condition is satisfied

life span of classical solutions to two dimensional fully nonlinear wave equations abstract in two space dimensional case we get the lower bound of the life span of classical solutions to the cauchy problem with small initial data for fully nonlinear wave equations of the form u f u d u d x d u in which f o 3 in a neighbourhood of 0 and u 3 f 0 0 0 0 for the purpose some refined estimates including strichartz estimates are needed in the paper

a vector field method for massless relativistic transport equations and applications abstract in this article we present a vector field method for the study of solutions to massless relativistic transport equations compared to 10 we remove the lorentz boosts of the commutation vector fields and we prove a new functional inequality in order to derive pointwise decay estimates on the solutions it makes our method more suitable for the study of systems coupling a massless vlasov equation with an equation which has a different speed of propagation we also believe that our approach can be adapted more easily to the context of curved backgrounds in the second part of this paper we apply our method in order to derive sharp decay estimates on the spherically symmetric small data solutions of the relativistic massless vlasov poisson system no compact support assumption is required on the data and in particular the initial decay in the velocity variable is optimal in order to compensate the slow decay rate of the particle density near the light cone we take advantage of the null structure of the equations

design of natural classification kernels using prior knowledge a new class of kernels has been designed to enhance the usability of prior knowledge prior knowledge is shown to improve the generalization ability of kernel algorithms for binary classification problems the prior knowledge is expressed in natural language via fuzzy rules first the concepts of a fuzzy rule base and prior confidence region are proposed to formulate the prior knowledge then the new kernels which are referred to as natural classification kernels ncks are represented by fuzzy equivalence relations based on the formulation of prior knowledge an nck is interpreted as a measure of similarities between samples it is proven that ncks have two desired properties 1 transitivity with respect to the triangular norms and 2 the ability to provide higher similarities to spatially closer samples from the same class using transitivity a large number of ncks may be directly obtained by means of triangular norms additionally the theoretical results show that the second property makes it possible for the support vector machine svm and convex hull separation algorithm to generalize from training samples to test samples in the prior confidence region finally some synthetic datasets and a benchmark dataset are employed to validate the proposed approach

global existence of radial solutions for general semilinear hyperbolic systems in 3d abstract we study the well posedness of radial solutions for general nonlinear hyperbolic systems in three dimensions we give a proof of the global existence of radial solutions for general semilinear hyperbolic systems in 3d under null condition with small scaling invariant w 2 1 r 3 data we obtain a bilinear estimate that is effective for hyperbolic systems which do not have any time decay it allows us to achieve the boundedness of the weighted bv norm of the radial solutions

decay for solutions of the wave equation on kerr exterior spacetimes iii the full subextremal case a m this paper concludes the series begun in m dafermos and i rodnianski decay for solutions of the wave equation on kerr exterior spacetimes i ii the cases jaj m or axisymmetry arxiv 1010 5132 providing the complete proof of denitive boundedness and decay results for the scalar wave equation on kerr backgrounds in the general subextremal jaj m case without symmetry assumptions the essential ideas of the proof together with explicit constructions of the most dicult mul

comments on a possibilistic approach to clustering in this comment we report a difficulty with the application of the possibilistic approach to fuzzy clustering pcm proposed by keller and krishnapuram 1993 in applying this algorithm we found that it has the undesirable tendency to produce coincidental clusters results illustrating this tendency are reported and a possible explanation for the pcm behavior is suggested

modified deflation algorithm for the solution of singular problems i a system of nonlinear algebraic equations abstract the behavior of the newton raphson method at the singular roots has been studied by a number of authors and the convergence of the newton raphson sequence has been shown to be linear in this paper a new method with symbolic and numerical manipulations termed the modified deflation algorithm is proposed for the singular root of a system of nonlinear algebraic equations the basic idea of the present method is to replace a part of the original equations by a set of new equations which pass through the singular root according to the method both convergency and accuracy can greatly be improved in addition it is often possible to obtain analytically the singular root from the new equations in order to show the effectiveness of the present method two illustrative examples are solved

secular growth estimates for hyperbolic systems abstract we consider a model problem for the secular growth which covers all the cases likely to happen in multi scales bkw expansions for nonlinear hyperbolic systems this model problem consists in studying the growth in time of the solution of a nonhomogeneous hyperbolic system whose source term is a product of various functions which solve homogeneous hyperbolic systems the secular growth is due to resonances that we try to control when this is not possible other tools such as decay properties or strichartz estimates must be used

asymptotics of radiation fields in asymptotically minkowski space we consider a non trapping n dimensional lorentzian manifold endowed with an end structure modeled on the radial compactification of minkowski space we find a full asymptotic expansion for tempered forward solutions of the wave equation in all asymptotic regimes the rates of decay seen in the asymptotic expansion are related to the resonances of a natural asymptotically hyperbolic problem on the northern cap of the compactification for small perturbations of minkowski space that fit into our framework our asymptotic expansions yield a rate of decay that improves on the klainerman sobolev estimates

the global cauchy problem for the critical non linear wave equation abstract we study the global cauchy problem for the non linear wave equation p 1 0 for the critical value p n 2 n 2 in space dimension n 3 we identify a weak space time integrability property stip of the solutions and prove that it is sufficient to ensure the uniqueness of weak solutions the global existence of finite energy solutions with the naturally associated stip and the global existence of regular solutions with some n dependent restrictions on the regularity for spherically symmetric solutions we prove that the previous crucial stip follows from the morawetz inequality actually in a much stronger form than necessary thereby proving that all the previous results hold in the spherically symmetric case

global solutions for incompressible viscoelastic fluids we prove the existence of both local and global smooth solutions to the cauchy problem in the whole space and the periodic problem in the n dimensional torus for the incompressible viscoelastic system of oldroyd b type in the case of near equilibrium initial data the results hold in both two and three dimensional spaces the results and methods presented in this paper are also valid for a wide range of elastic complex fluids such as magnetohydrodynamics liquid crystals and mixture problems

global solutions of nonlinear wave equations in time dependent inhomogeneous media we consider the problem of small data global existence for a class of semilinear wave equations with null condition on a lorentzian background mathbb r 3 1 g with a time dependent metric g coinciding with the minkowski metric outside the cylinder t x x leq r we show that the small data global existence result can be reduced to two integrated local energy estimates and demonstrate that these estimates work in the particular case when g is merely c1 close to the minkowski metric one of the novel aspects of this work is that it applies to equations on backgrounds which do not settle to any particular stationary metric

asymptotics of scalar waves on long range asymptotically minkowski spaces abstract we show the existence of the full compound asymptotics of solutions to the scalar wave equation on long range non trapping lorentzian manifolds modeled on the radial compactification of minkowski space in particular we show that there is a joint asymptotic expansion at null and timelike infinity for forward solutions of the inhomogeneous equation in two appendices we show how these results apply to certain spacetimes whose null infinity is modeled on that of the kerr family in these cases the leading order logarithmic term in our asymptotic expansions at null infinity is shown to be nonzero

global existence for three dimensional incompressible isotropic elastodynamics via the incompressible limit the existence of global in time classical solutions to the cauchy problem for incompressible nonlinear isotropic elastodynamics for small initial displacements is proved solutions are constructed via approximation by slightly compressible materials the energy for the approximate solutions remains uniformly bounded on a time scale that goes to infinity as the material approaches incompressibility a necessary component to the long time existence of the approximating solution is a null or linear degeneracy condition inherent in the isotropic case which limits the quadratic interaction of the shear waves the proof combines energy and decay estimates based on commuting vector fields and a compactness argument 2004 wiley periodicals inc

almost global wellposedness of the 2 d full water wave problem we consider the problem of global in time existence and uniqueness of solutions of the 2 d infinite depth full water wave equation it is known that this equation has a solution for a time period 0 t e for initial data of the form e where t depends only on in this paper we show that for such data there exists a unique solution for a time period 0 e t e this is achieved by better understandings of the nature of the nonlinearity of the full water wave equation

estimating the support of a high dimensional distribution suppose you are given some data set drawn from an underlying probability distribution p and you want to estimate a simple subset s of input space such that the probability that a test point drawn from p lies outside of s equals some a priori specified value between 0 and 1 we propose a method to approach this problem by trying to estimate a function f that is positive on s and negative on the complement the functional form of f is given by a kernel expansion in terms of a potentially small subset of the training data it is regularized by controlling the length of the weight vector in an associated feature space the expansion coefficients are found by solving a quadratic programming problem which we do by carrying out sequential optimization over pairs of input patterns we also provide a theoretical analysis of the statistical performance of our algorithm the algorithm is a natural extension of the support vector algorithm to the case of unlabeled data

life span of small solutions to a system of wave equations abstract we study the cauchy problem with small initial data for a system of semilinear wave equations u v q v t u p in n dimensional space when n 2 we prove that blow up can occur for arbitrarily small data if p q lies below a curve in the p q plane on the other hand we show a global existence result for n 3 which asserts that a portion of the curve is in fact the borderline between global in time existence and finite time blow up we also estimate the maximal existence time and get its upper bound which is sharp at least for n p q 2 2 2 and 3 2 2

uniform decay estimates and the lorentz invariance of the classical wave equation on etudie le comportement asymptotique des solutions de u 0 ou t 2 1 2 n 2 pour des conditions initiales u 0 u t g x en t 0 avec g reguliere a support compact dans r n

global existence of small solutions to nonlinear evolution equations abstract we study the global existence and asymptotic behaviour of small solutions of a large class of nonlinear partial differential equations if the nonlinear terms are of high degree the solutions will be asymptotic to solutions of the linear equation

input space versus feature space in kernel based methods this paper collects some ideas targeted at advancing our understanding of the feature spaces associated with support vector sv kernel functions we first discuss the geometry of feature space in particular we review what is known about the shape of the image of input space under the feature space map and how this influences the capacity of sv methods following this we describe how the metric governing the intrinsic geometry of the mapped surface can be computed in terms of the kernel using the example of the class of inhomogeneous polynomial kernels which are often used in sv pattern recognition we then discuss the connection between feature space and input space by dealing with the question of how one can given some vector in feature space find a preimage exact or approximate in input space we describe algorithms to tackle this issue and show their utility in two applications of kernel methods first we use it to reduce the computational complexity of sv decision functions second we combine it with the kernel pca algorithm thereby constructing a nonlinear statistical denoising technique which is shown to perform well on real world data