the aw rascle traffic model with locally constrained flow abstract we consider solutions of the aw rascle model for traffic flow fulfilling a constraint on the flux at x 0 two different kinds of solutions are proposed at x 0 the first one conserves both the number of vehicles and the generalized momentum while the second one conserves only the number of cars we study the invariant domains for these solutions and we compare the two riemann solvers in terms of total variation of relevant quantities finally we construct ad hoc finite volume numerical schemes to compute these solutions

invariant region preserving dg methods for multi dimensional hyperbolic conservation law systems with an application to compressible euler equations abstract an invariant region preserving irp limiter for multi dimensional hyperbolic conservation law systems is introduced as long as the system admits a global invariant region which is a convex set in the phase space it is shown that the order of approximation accuracy is not destroyed by the irp limiter provided the cell average is away from the boundary of the convex set moreover this limiter is explicit and easy for computer implementation a generic algorithm incorporating the irp limiter is presented for high order finite volume type schemes for arbitrarily high order discontinuous galerkin dg schemes to hyperbolic conservation law systems sufficient conditions are obtained for cell averages to remain in the invariant region provided the projected one dimensional system shares the same invariant region as the full multi dimensional hyperbolic system does the general results are then applied to both one and two dimensional compressible euler equations so to obtain high order irp dg schemes numerical experiments are provided to validate the proven properties of the irp limiter and the performance of irp dg schemes for compressible euler equations

crises sudden changes in chaotic attractors and transient chaos abstract the occurrence of sudden qualitative changes of chaotic dynamics as a parameter is varied is discussed and illustrated it is shown that such changes may result from the collision of an unstable periodic orbit and a coexisting chaotic attractor we call such collisions crises phenomena associated with crises include sudden changes in the size of chaotic attractors sudden appearances of chaotic attractors a possible route to chaos and sudden destructions of chaotic attractors and their basins this paper presents examples illustrating that crisis events are prevalent in many circumstances and systems and that just past a crisis certain characteristic statistical behavior whose type depends on the type of crisis occurs in particular the phenomenon of chaotic transients is investigated the examples discussed illustrate crises in progressively higher dimension and include the one dimensional quadratic map the two dimensional henon map systems of ordinary differential equations in three dimensions and a three dimensional map in the case of our study of the three dimensional map a new route to chaos is proposed which is possible only in invertible maps or flows of dimension at least three or four respectively based on the examples presented the following conjecture is proposed almost all sudden changes in the size of chaotic attractors and almost all sudden destruction or creations of chaotic attractors and their basins are due to crises

convergence of approximate solutions of the cauchy problem for a 2x2 nonstrictly hyperbolic system of conservation laws a convergence theorem for the vanishing viscosity method and for the lax friedrichs schemes applied to a nonstrictly hyperbolic and nongenuinely nonlinear system is established using the theory of compensated compactness we prove convergence of a subsequence in the strong topology

complex dynamics in the 1 3 spatial resonance abstract the interaction between two steady state bifurcations with spatial wave numbers in the ratio 1 3 is considered periodic boundary conditions are assumed the resulting o 2 equivariant normal form truncated at third order exhibits a number of global bifurcations that may result in complex dynamics the origin of this behavior is elucidated with the help of careful numerical simulations and analysis of appropriate return maps the results generalize to other 1 n resonances

on the numerical approximation of the rotation number abstract the starting point of this paper is a polygonal approximation of an invariant curve of a map using this polygonal approximation an approximation for the circle map the restriction of the map to the invariant curve is obtained the rotation number of the circle map is then approximated by the rotation number of the approximated circle map the error in the obtained approximate rotation number is discussed and related to the error in the polygonal approximation of the invariant curve simple algorithms for the approximation of the rotation number are described a numerical example illustrates the theory

dynamics of a parametrically excited double pendulum abstract the 2 2 mode interaction of a parametrically excited double pendulum is explored in the excitation frequency excitation amplitude plane to determine the bifurcation structure at small amplitudes of oscillation the method of averaging combined with centre manifold reduction is used the full equations are solved numerically to extend the bifurcation set to larger amplitudes of response numerical centre manifold reduction is employed to derive two maps valid near two multiple bifurcation points which organise the dynamical phenomena of the mode interaction region iteration of these maps shows the existence of global bifurcations these results are discussed in the light of numerical integrations which show that a whole range of interesting behaviour occurs including torus doubling torus gluing and chaos

error bounds for finite difference approximations for a class of nonlinear parabolic systems in this paper we establish error bounds for a finite difference approximation to solutions of certain parabolic systems of the form t f po u we assume that the cauchy data is of class bv and we show that the sup norm of the error is bounded by o axiln axi at positive times

granularity selection for cross validation of svm abstract granularity selection is fundamental to granular computing cross validation cv is widely adopted for model selection where each fold of data set of cv can be considered as an information granule and the larger the number of the folds is the smaller the granularity of each fold is therefore for cv granularity selection is equal to the selection of the number of folds in this paper we explore the granularity selection for cv of support vector machine svm we first use the huber loss to smooth the hinge loss used in svm and to approximate cv of svm then we derive a tight upper bound of the discrepancy between the original and the approximate cv with a high convergence rate finally based on this derived tight bound we present a granularity selection criterion for trading off the accuracy and time cost experimental results demonstrate that the approximate cv with the granularity selection criterion gives the similar accuracies as the traditional cv and meanwhile significantly improves the efficiency

robustness and regularization of support vector machines we consider regularized support vector machines svms and show that they are precisely equivalent to a new robust optimization formulation we show that this equivalence of robust optimization and regularization has implications for both algorithms and analysis in terms of algorithms the equivalence suggests more general svm like algorithms for classification that explicitly build in protection to noise and at the same time control overfitting on the analysis front the equivalence of robustness and regularization provides a robust optimization interpretation for the success of regularized svms we use this new robustness interpretation of svms to give a new proof of consistency of kernelized svms thus establishing robustness as the reason regularized svms generalize well

observations of order and chaos in nonlinear systems experiments on nonlinear electrical oscillators the belousov zhabotinskii reaction rayleigh benard convection and couette taylor flow have revealed several common routes to chaos that have also been found in numerical studies of models with a few degrees of freedom experimental results are presented illustrating the following transition sequences period doubling and the u sequence intermittency the periodic quasiperiodic chaotic sequence frequency locking and an alternating periodic chaotic sequence

entropy flux splittings for hyperbolic conservation laws part i general framework a general framework is proposed for the derivation and analysis of flux splittings and the corresponding flux splitting schemes for systems of conservation laws endowed with a strictly convex entropy the approach leads to several new properties of the existing flux splittings and to a method for the construction of entropy flux splittings for general situations a large family of genuine entropy flux splittings is derived for several significant examples the scalar conservation laws the p system and the euler system of isentropic gas dynamics in particular for the isentropic euler system we obtain a family of splittings that satisfy the entropy inequality associated with the mechanical energy for this system it is proved that there exists a unique genuine entropy flux splitting that satisfies all of the entropy inequalities which is also the unique diagonalizable splitting this splitting can be also derived by the so called kinetic formulation simple and useful difference schemes are derived from the flux splittings for hyperbolic systems such entropy flux splitting schemes are shown to satisfy a discrete cell entropy inequality for the diagonalizable splitting schemes an a priori l estimate is provided by applying the principle of bounded invariant regions the convergence of entropy flux splitting schemes is proved for the 2 2 systems of conservation laws and the isentropic euler system 1995 john wiley sons inc

a class of convergent finite difference schemes for certain nonlinear parabolic systems on considere le systeme des equations v t u x ev xx u t p v x u xx x t rxr avec les donnees initiales v u x 0 v 0 x u 0 x x r

invariant domains and second order continuous finite element approximation for scalar conservation equations we propose a technique for approximating nonlinear scalar conservation equations that uses continuous finite elements and is formally at least second order accurate in space and maximum principle preserving the method is explicit in time uses unstructured continuous finite elements in space and works in any space dimension the stability and accuracy of the method are achieved by adapting the artificial viscosity

universal properties of the transition from quasi periodicity to chaos in dissipative systems abstract an exact renormalization group transformation is developed for dissipative systems which describes how the transition to chaos may occur in a continuous and universal manner if the frequency ratio in the quasi periodic regime is held at a fixed irrational value our approach is a natural extension of k a m theory to strong coupling most of our analysis is for analytic circle maps we have found a strong coupling fixed point where invertibility is lost which describes the universal features of the transition to chaos we find numerically that any two such critical maps with the same winding number are c 1 conjugate it follows that the low frequency peaks in an experimental spectrum are universal and we determine how their envelope scales with frequency when the winding number has a periodic continued fraction our renormalization transform has a fixed point and spectra are self similar in addition for a set of non periodic winding numbers with full measure our renormalization transformation yields an ergodic trajectory in a sub space of all critical maps physically one finds singular and universal spectra that do not scale

quasiperiodicity in dissipative systems a renormalization group analysis abstract dynamical systems with quasiperiodic behavior i e two incommensurate frequencies may be studied via discrete maps which show smooth continuous invariant curves with irrational winding number in this paper these curves are followed using renormalization group techniques which are applied to a one dimensional system circle and also to an area contracting map of an annulus two fixed points are found representing different types of universal behavior a trivial fixed point for smooth motion and a nontrivial fixed point the latter representsthe incipient breakup of a quasiperiodic motion with frequency ratio the golden mean into a more chaotic flow fixed point functions are determined numerically and via an e expansion and eigenvalues are calculated

normal forms for driven surface waves boundary conditions symmetry and genericity abstract the parametric excitation of standing waves in a vertically oscillated fluid can be studied by analyzing the period doubling bifurcation in the associated stroboscopic map when the fluid container has symmetry the stroboscopic map will exhibit this symmetry and this in turn significantly affects the structure of the bifurcation problem for both square and circular containers the normal forms appropriate to the reduced maps for the critical mode amplitudes are discussed the experiments of simonelli and gollub with square containers strongly suggest that the effective symmetry of the problem is larger than the obvious geometric symmetry of the boundaries for the ideal fluid model of benjamin and ursell with neumann boundary conditions nbc one can show that the problem does indeed have a larger symmetry because it embeds in a model on a larger domain with periodic boundary conditions pbc the original nbc are then realized as a symmetry constraint on the solutions of this extended model the additional symmetries introduced in this way lead to a large number of effects that would otherwise be non generic including critical eigenspaces carrying reducible representations of and reduced maps on these eigenspaces that have symmetries larger than these extra symmetries relate in phase and out of phase mixed modes and lead to new branches to pure modes the case of circular geometry is also discussed as it does not allow this extension to pbc and thus provides an instructive contrast with the subtleties found for square geometry

merging of resonance tongues abstract this paper considers the resonance behavior found near a saddle node hopf bifurcation in a flow the global structure of resonance tongues is examined and it is shown that in the absence of certain symmetries different resonance tongues can merge with one another the position and shape of the resonance tongues are seen to be related to the presence of homoclinic and heteroclinic bifurcations of two saddle foci the implications of these results for the normal form analysis of the saddle node hopf bifurcation are discussed

a heterogeneous multiclass traffic flow model with creeping a heterogeneous traffic model with two vehicle classes is developed to capture overtaking and creeping in highly heterogeneous traffic flows creeping is a special case of overtaking that occurs when small vehicles continue to advance in congestion even though larger vehicles have completely stopped to motivate the new model it is shown that a two class homogeneous multiclass model is equivalent to a class of second order models originally proposed by aw rascle and zhang arz based on the properties of the arz model homogeneous models do not allow overtaking or creeping the new creeping model is a phase transition model which applies a system of conservation laws in the noncreeping phase and a system equivalent to a scalar model in the creeping phase the solution to the riemann problem is obtained by investigating the elementary waves particularly for the cases when one vehicle class is absent as well as in the presence of a phase transition based on the proposed riemann solver the solution t

two frequency forced faraday waves weakly damped modes and pattern selection abstract recent experiments a kudrolli b pier j p gollub physica d 123 1998 99 111 on two frequency parametrically excited surface waves produced an intriguing superlattice wave pattern near a codimension two bifurcation point where both subharmonic and harmonic waves onset simultaneously but with different spatial wave numbers the superlattice pattern is synchronous with the forcing spatially periodic on a large hexagonal lattice and exhibits small scale triangular structure similar patterns have been shown to exist as primary solution branches of a generic 12 dimensional d 6 t 2 equivariant bifurcation problem and may be stable if the nonlinear coefficients of the bifurcation problem satisfy certain inequalities m silber m r e proctor phys rev lett 81 1998 2450 2453 here we use the spatial and temporal symmetries of the problem to argue that weakly damped harmonic waves may be critical to understanding the stabilization of this pattern in the faraday system we illustrate this mechanism by considering the equations developed by zhang and vinals j fluid mech 336 1997 301 330 for small amplitude weakly damped surface waves on a semi infinite fluid layer we compute the relevant nonlinear coefficients in the bifurcation equations describing the onset of patterns for excitation frequency ratios of 2 3 and 6 7 for the 2 3 case we show that there is a fundamental difference in the pattern selection problems for subharmonic and harmonic instabilities near the codimension two point also we find that the 6 7 case is significantly different from the 2 3 case due to the presence of additional weakly damped harmonic modes these additional harmonic modes can result in a stabilization of the superpatterns

traffic flow modelling with junctions abstract motivated by the modelling of a roundabout we are led to study the traffic on a road with points of entry and exit in this note we would like to describe the modellisation of a junction and solve the riemann problem for such a model more precisely between each point of discontinuity we use a multi class extension of the lwr model to describe the evolution of the density of the vehicles the multi class approach being used in order to distinguish the vehicles after their origin and destination then we treat the points of entry and exit thanks to special boundary conditions that give bounds on the flows of the different types of vehicles in the case of the one t road we obtain a result of existence and uniqueness this first step allows us to obtain a similar result for the n t road we describe these results and also some properties of the obtained solutions in order to see how long this model is valid

the error in spatial truncation for systems of parabolic conservation laws on etudie le comportement de la solution de u t du xx f u x u o x u 0 x l u t l u ou t 0 et x l l on obtient des resultats de decroissance relatifs aux normes de la difference entre la solution pour l infini et pour l fini

a 2 phase traffic model based on a speed bound we extend the classical lighthill whitham richards lwr traffic model allowing different maximal speeds for different vehicles then we add a uniform bound on the traffic speed the result presented in this paper is a new macroscopic model displaying two phases based on a nonsmooth 2 times2 system of conservation laws this model is compared with other models of the same type in the current literature as well as with a kinetic one moreover we establish a rigorous connection between a microscopic follow the leader model based on ordinary differential equations and this macroscopic continuum model

hopf saddle node bifurcation for fixed points of 3d diffeomorphisms analysis of a resonance bubble the dynamics near a hopf saddle node bifurcation of fixed points of diffeomorphisms is analysed by means of a case study a two parameter model map g is constructed such that at the central bifurcation the derivative has two complex conjugate eigenvalues of modulus one and one real eigenvalue equal to 1 to investigate the effect of resonances the complex eigenvalues are selected to have a 1 5 resonance it is shown that near the origin of the parameter space the family g has two secondary hopf saddle node bifurcations of period five points a cone like structure exists in the neighbourhood formed by two surfaces of saddle node and a surface of hopf bifurcations quasi periodic bifurcations of an invariant circle forming a frayed boundary are numerically shown to occur in model g along such cantor like boundary an intricate bifurcation structure is detected near a 1 5 resonance gap subordinate quasi periodic bifurcations are found nearby suggesting the occurrence of a cascade of quasi periodic bifurcations c 2008 elsevier b v all rights reserved

maps of convex sets and invariant regions for finite difference systems of conservation laws general results about maps of convex sets in r n are proved we outline their extensions to an infinite dimensional context such extensions have applications in nonlinear analysis such as in the study of the invariance of convex sets under nonlinear maps here we explore applications only in the finite dimensional context more specifically we apply the general results to the problem of finding sufficient conditions for a region of the state space to be globally or locally invariant under finite difference schemes applied to systems of conservation laws in several space variables in particular we establish a final characterization of the invariant regions under the lax friedrichs scheme and also give sufficient conditions for the local invariance further we give sufficient conditions for the global and local invariance of regions under flux splitting finite difference schemes an example of the multi dimensional euler equations for non isentropic gas dynamics is discussed

nonparametric quantile estimation in regression the desired estimate of y x is not always given by a conditional mean although this is most common sometimes one wants to obtain a good estimate that satisfies the property that a proportion of y x will be below the estimate for 0 5 this is an estimate of the median what might be called median regression is subsumed under the term quantile regression we present a nonparametric version of a quantile estimator which can be obtained by solving a simple quadratic programming problem and provide uniform convergence statements and bounds on the quantile property of our estimator experimental results show the feasibility of the approach and competitiveness of our method with existing ones we discuss several types of extensions including an approach to solve the quantile crossing problems as well as a method to incorporate prior qualitative knowledge such as monotonicity constraints

well balanced second order approximation of the shallow water equation with continuous finite elements this paper investigates a first order and a second order approximation technique for the shallow water equation with topography using continuous finite elements both methods are explicit in time and are shown to be well balanced the first order method is invariant domain preserving and satisfies local entropy inequalities when the bottom is flat both methods are positivity preserving both techniques are parameter free work well in the presence of dry states and can be made high order in time by using strong stability preserving time stepping algorithms

robust asymptotic statistics contents preface notation chapter 1 von mises functionals chapter 2 log likelihoods chapter 3 asymptotic statistics chapter 4 nonparametric statistics chapter 5 optimal influence curves chapter 6 stable constructions chapter 7 robust regression appendix a weak convergence of measures appendix b some functional analysis appendix c complements bibliography index

fast estimation from above of the maximum wave speed in the riemann problem for the euler equations this paper is concerned with the construction of a fast algorithm for computing the maximum speed of propagation in the riemann solution for the euler system of gas dynamics with the co volume equation of state the novelty in the algorithm is that it stops when a guaranteed upper bound for the maximum speed is reached with a prescribed accuracy the convergence rate of the algorithm is cubic and the bound is guaranteed for gasses with the co volume equation of state and the heat capacity ratio in the range 1 5 3

invariant domain preserving discretization independent schemes and convex limiting for hyperbolic systems abstract we introduce an approximation technique for nonlinear hyperbolic systems with sources that is invariant domain preserving the method is discretization independent provided elementary symmetry and skew symmetry properties are satisfied by the scheme the method is formally first order accurate in space then we introduce a series of higher order methods when these methods violate the invariant domain properties they are corrected by a limiting technique that we call convex limiting after limiting the resulting methods satisfy all the invariant domain properties that are imposed by the user see theorem 7 24 and is formally high order accurate the two key novelties are that i limiting is done by enforcing bounds on quasiconcave functionals ii the bounds that are enforced on the solution at each time step are necessarily satisfied by the low order approximation

period doubling mode interactions with circular symmetry abstract a two parameter analysis of the interaction between two period doubling modes with azimuthal wavenumbers k and l l k 1 is carried out for systems with circular symmetry the problem is formulated in terms of an o 2 equivariant map on c 2 in the generic case all primary secondary and tertiary solution branches and their stability properties are classified the results depend on whether k l is odd or even when one mode bifurcates subcritically and the other supercritically the pure mode branches lose stability to a branch of reflection symmetric mixed modes which in turn can undergo a tertiary hopf bifurcation to a quasiperiodic reflection symmetric pattern we conjecture that this invariant circle can break down with increasing amplitude to produce reflection symmetric chaos additional phase instabilities may occur in which the reflection symmetry is broken and the resulting pattern drifts either clockwise or counterclockwise the results explain a number of observations by ciliberto and gollub on parametrically excited surface waves in circular geometry and imply several new predictions for such experiments in an appendix the theory is compared with previous attempts to model the experiments

learning with the maximum correntropy criterion induced losses for regression within the statistical learning framework this paper studies the regression model associated with the correntropy induced losses the correntropy as a similarity measure has been frequently employed in signal processing and pattern recognition motivated by its empirical successes this paper aims at presenting some theoretical understanding towards the maximum correntropy criterion in regression problems our focus in this paper is two fold first we are concerned with the connections between the regression model associated with the correntropy induced loss and the least squares regression model second we study its convergence property a learning theory analysis which is centered around the above two aspects is conducted from our analysis we see that the scale parameter in the loss function balances the convergence rates of the regression model and its robustness we then make some efforts to sketch a general view on robust loss functions when being applied into the learning for regression problems numerical experiments are also implemented to verify the effectiveness of the model

chaotic dynamics of a bouncing ball abstract a detailed study of a mapping on a two dimensional manifold is made the mapping describes a system subject to periodic forcing in particular an imperfectly elastic ball bouncing on a vibrating platform quasiperiodic motion on a one dimensional manifold is proven and observed numerically at low forcing while at higher forcing smale horseshoes are present we examine the evolution of the attracting set with changing parameter spatial structure is oganised by fixed points of the mapping and sudden changes occur by crises a new type of chaos in which a trajectory alternates between two distinct chaotic regions is described and explained in terms of manifold collisions throughout we are concerned to examine the behaviour of lyapunov exponents typical behaviour of lyapunov exponents in the quasiperiodic regime under the influence of external noise is discussed at higher forcing a certain symmetry of the attractor allows an analytic expression for the exponents to be given

fast cross validation for kernel based algorithms cross validation cv is a widely adopted approach for selecting the optimal model however the computation of empirical cross validation error cve has high complexity due to multiple times of learner training in this paper we develop a novel approximation theory of cve and present an approximate approach to cv based on the bouligand influence function bif for kernel based algorithms we first represent the bif and higher order bifs in taylor expansions and approximate cv via the taylor expansions we then derive an upper bound of the discrepancy between the original and approximate cv furthermore we provide a novel computing method to calculate the bif for general distribution and evaluate bif criterion for sample distribution to approximate cv the proposed approximate cv requires training on the full data set only once and is suitable for a wide variety of kernel based algorithms experimental results demonstrate that the proposed approximate cv is sound and effective

on robustness properties of convex risk minimization methods for pattern recognition the paper brings together methods from two disciplines machine learning theory and robust statistics we argue that robustness is an important aspect and we show that many existing machine learning methods based on the convex risk minimization principle have besides other good properties also the advantage of being robust robustness properties of machine learning methods based on convex risk minimization are investigated for the problem of pattern recognition assumptions are given for the existence of the influence function of the classifiers and for bounds on the influence function kernel logistic regression support vector machines least squares and the adaboost loss function are treated as special cases some results on the robustness of such methods are also obtained for the sensitivity curve and the maxbias which are two other robustness criteria a sensitivity analysis of the support vector machine is given

bifurcations from an invariant circle for two parameter families of maps of the plane a computer assisted study we consider a two parameter family of maps of the plane to itself each map has a fixed point in the first quadrant and is a diffeomorphism in a neighborhood of this point for certain parameter values there is a hopf bifurcation to an invariant circle which is smooth for parameter values in a neighborhood of the bifurcation point however computer simulations show that the corresponding invariant set fails to be even topologically a circle for parameter values far from the bifurcation point this paper is an attempt to elucidate some of the mechanisms involved in this loss of smoothness and alteration of topological type

bouligand derivatives and robustness of support vector machines for regression we investigate robustness properties for a broad class of support vector machines with non smooth loss functions these kernel methods are inspired by convex risk minimization in infinite dimensional hilbert spaces leading examples are the support vector machine based on the e insensitive loss function and kernel based quantile regression based on the pinball loss function firstly we propose with the bouligand influence function bif a modification of f r hampel s influence function the bif has the advantage of being positive homogeneous which is in general not true for hampel s influence function secondly we show that many support vector machines based on a lipschitz continuous loss function and a bounded kernel have a bounded bif and are thus robust in the sense of robust statistics based on influence functions

lack of bv bounds for approximate solutions to the p system with large data we consider front tracking approximate solutions to the p system of isentropic gas dynamics at interaction times the outgoing wave fronts have the same strength as in the exact solution of the riemann problem but some error is allowed in their speed for large bv initial data we construct examples showing that the total variation of these approximate solutions can become arbitrarily large or even blow up in finite time this happens even if the density of the gas remains uniformly positive

well posedness of balance laws with boundary consider the initial boundary value problem for a temple system of balance laws aim of this paper is to prove the well posedness of this problem for large times and without requiring the total variation of the initial data be small

local convergence of quasi newton methods for b differentiable equations we study local convergence of quasi newton methods for solving systems of nonlinear equations defined by b differentiable functions we extend the classical linear and superlinear convergence results for general quasi newton methods as well as for broyden s method we also show how broyden s method may be applied to nonlinear complementarity problems and illustrate its computational performance on two small examples

simplified godunov schemes for 2 x 2 systems of conservation laws some godunov type schemes for solving isentropic gas dynamic equations are presented after analyzing a one intermediary state scheme and its properties we prove consistency with entropy condition for time continuous limit solutions of a simplified godunov scheme by techniques due to osher we prove also similar results for high order versions of these numerical schemes

fourier methods for quasi periodic oscillations quasi periodic oscillations and invariant tori play an important role in the study of forced or coupled oscillators this paper presents two new numerical methods for the investigation of quasi periodic oscillations both algorithms can be regarded as generalizations of the averaging and the harmonic spectral balance methods the algorithms are easy to implement and require only minimal a priori knowledge of the system most importantly the methods do not depend on an a priori co ordinate transformation the methods are applied to a number of illustrative examples from non linear electrical engineering and the results show that the methods are efficient and reliable in addition these examples show that the presented algorithms can also continue through regions of sub harmonic phase locked resonance even though they are designed only for the quasi periodic case copyright 2006 john wiley sons ltd

amplitude equations for systems with competing instabilities we describe how to derive ordinary differential equations to predict the time dependence of a system governed by partial differential equations when that system is near to the polycritical condition for the onset of several instabilities the method is illustrated for the general case of two competing instabilities and then a specific physical example of this case is worked out in detail the basic idea is to extend the krylov bogolyubov mitropolsky method of nonlinear mechanics to bifurcation problems

invariant regions for systems of conservation laws on decrit des conditions necessaires et suffisantes pour qu une region dans r n soit invariante pour des solutions du systeme de n lois de conservations u t f u x o

on the influence of the kernel on the consistency of support vector machines in this article we study the generalization abilities of several classifiers of support vector machine svm type using a certain class of kernels that we call universal it is shown that the soft margin algorithms with universal kernels are consistent for a large class of classification problems including some kind of noisy tasks provided that the regularization parameter is chosen well in particular we derive a simple sufficient condition for this parameter in the case of gaussian rbf kernels on the one hand our considerations are based on an investigation of an approximation property the so called universality of the used kernels that ensures that all continuous functions can be approximated by certain kernel expressions this approximation property also gives a new insight into the role of kernels in these and other algorithms on the other hand the results are achieved by a precise study of the underlying optimization problems of the classifiers furthermore we show consistency for the maximal margin classifier as well as for the soft margin svm s in the presence of large margins in this case it turns out that also constant regularization parameters ensure consistency for the soft margin svm s finally we prove that even for simple noise free classification problems svm s with polynomial kernels can behave arbitrarily badly