polynomial lienard equations near infinity abstract we study the behaviour near infinity of the generalized lienard equations x y y m k 0 a k x k y n k 0 b k x k providing a complete classification using poincare compactification as well as poincare lyapunov compactification we show that all the necessary information is contained in a m and b n except for the so called center focus problem occuring in case m 2 n 1 with m n odd and where the behaviour also depends on the value of the other coefficients a i and b j we also indicate how to take care about obtaining uniform information near infinity

approximation of cahn hilliard diffuse interface models using parallel adaptive mesh refinement and coarsening with c1 elements a variational formulation and c1 finite element scheme with adaptive mesh refinement and coarsening are developed for phase separation processes described by the cahn hilliard diffuse interface model of transport in a mixture or alloy the adaptive scheme is guided by a laplacian jump indicator based on the corresponding term arising from the weak formulation of the fourth order non linear problem and is implemented in a parallel solution framework it is then applied to resolve complex evolving interfacial solution behavior for 2d and 3d simulations of the classic spinodal decomposition problem from a random initial mixture and to other phase transformation applications of interest simulation results and adaptive performance are discussed the scheme permits efficient robust multiscale resolution and interface characterization copyright 2008 john wiley sons ltd

spline space and its b splines on an n 1 direction mesh in r n it is well known that splines play an important role in many fields especially their close relationship with wavelets makes them have more widespread applications in numerous scientific and engineering domains univariate and bivariate splines have been well studied and lots of results have been obtained because of the intrinsic difficulty between bivariate case and higher dimension three or more dimensions settings the study of splines on higher dimensions are very limited for example the study of the bivariate splines on a three direction mesh triangulation has obtained many important and excellent results but almost all of those results have no analog generalization to higher dimensions in this paper we will study the higher dimension splines defined on n 1 mesh simplical partitions which is the analog of bivariate splines on three mesh triangulations we have also pointed out many interesting differences between bivariate splines and higher dimensional cases our main results are that similar to bivariate and trivariate cases a necessary and sufficient condition for skr to contain a b spline is k r 1 n 1 for r 1 being odd and k r n 1 1 for r 0 being even

on the approximation power of bivariate quadratic c splines abstract in this paper we investigate the approximation power of local bivariate quadratic c1 quasi interpolating q i spline operators with a four directional mesh in particular we show that they can approximate a real function and its partial derivatives up to an optimal order and we derive local and global upper bounds both for the errors and for the spline partial derivatives in the case the spline is more differentiable than the function then such general results are applied to prove new properties of two interesting q i spline operators proposed and partially studied in chui and wang sci sinica xxvii 1984 1129 1142

symmetric box splines on root lattices root lattices are efficient sampling lattices for reconstructing isotropic signals in arbitrary dimensions due to their highly symmetric structure one root lattice the cartesian grid is almost exclusively used since it matches the coordinate grid but it is less efficient than other root lattices box splines on the other hand generalize tensor product b splines by allowing non cartesian directions they provide in any number of dimensions higher order reconstructions of fields often of higher efficiency than tensored b splines but on non cartesian lattices such as the bcc body centered cubic or the fcc face centered cubic lattice only some box splines and then only up to dimension three have been investigated this paper derives and completely characterizes efficient symmetric box spline reconstruction filters on all irreducible root lattices that exist in any number of dimensionsn 2 n 3 for d n and d n lattices in all cases box splines are constructed by convolution using the lattice directions generalizing the known constructions in two and three variables for each box spline we document the basic properties for computational use the polynomial degree the continuity the linear independence of shifts on the lattice and optimal quasi interpolants for fast approximation of fields

asymptotic normality with small relative errors of posterior probabilities of half spaces some examples of half spaces of interest in parameter spaces are in a clinical trial of a treatment versus placebo the half spaces where the treatment is a helpful or b harmful thus one may not only want to test the hypothesis that the treatment c makes no difference but to assign posterior probabilities to a b and c under conditions as unrestrictive as possible on the choice of prior probabilities e g dudley and haughton 2001 more generally we have in mind applications to model selections as in the bic criterion of schwarz 1978 and its extensions poskitt 1987 haughton 1988 specifically to one sided models and multiple data sets dudley and haughton 1997

quadratic lienard equations with quadratic damping beijing univ dept math beijing 100871 peoples r china beijing univ math inst beijing 100871 peoples r china dumortier f limburgs univ ctr univ campus b 3590 diepenbeek belgium

on bayesian supremum norm contraction rates building on ideas from castillo and nickl 6 a method is pro vided to study nonparametric bayesian posterior convergence rates when strong measures of distances such as the sup norm are considered in particular we show that likelihood methods can achieve optimal minimax sup norm rates in density estimation on the unit interval the introduced methodology is used to prove that commonly used families of prior distri butions on densities namely log density priors and dyadic random density histograms can indeed achieve optimal sup norm rates of convergence new results are also derived in the gaussian white noise model as a further il lustration of the presented techniques ams 2000 subject classications primary 62g20 secondary 62g05 62g07

compactification and desingularization of spaces of polynomial lienard equations abstract the paper deals with polynomial lienard equations of type m n i e planar vector fields associated to a scalar second order differential equation x f x x g x 0 with f and g polynomials of respective degree m and n it is shown that besides compactifying the phase plane or the lienard plane one can also compactify and desingularize the space of lienard equations of type m n for each m n separately by adding both singular perturbation problems and hamiltonian perturbation problems

macro elements and stable local bases for splines on powell sabin triangulations macro elements of arbitrary smoothness are constructed on powell sabin triangle splits these elements are useful for solving boundary value problems and for interpolation of hermite data it is shown that they are optimal with respect to spline degree and we believe they are also optimal with respect to the number of degrees of freedom the construction provides local bases for certain superspline spaces defined over powell sabin refinements these bases are shown to be stable as a function of the smallest angle in the triangulation which in turn implies that the associated spline spaces have optimal order approximation power

on robust analysis of a normal location parameter pericchi and smith considered a normal location parameter problem with double exponential and student t prior distributions these two prior distributions both belong to the class of scale mixtures of normal distributions and are useful in providing a robust analysis of the normal location parameter problem in this paper we extend the analysis to other scale mixtures of normal distributions such as the exponential power and the symmetric stable distributions

slow fast bogdanov takens bifurcations abstract in this paper we study perturbations from planar vector fields having a line of zeros and representing a singular limit of bogdanov takens bt bifurcations we introduce among other precise definitions the notion of slow fast bt bifurcation and we provide a complete study of the bifurcation diagram and the related phase portraits based on geometric singular perturbation theory including blow up we get results that are valid on a uniform neighborhood both in parameter space and in the phase plane

a note on bounded influence in bayesian analysis summary let x be a single observation from a distribution having unknown location parameter 0 dawid 1973 provided sufficient conditions for the posterior distribution of 0 to approach the prior distribution as x tends to infinity so that an outlier has bounded and vanishing influence on the posterior distribution we present a result closely related to dawid s theorem this enables us to consider priors and likelihoods for the location problem which have bounded but nonvanishing influence on posterior moments examples are given

configurations of limit cycles in lienard equations abstract we show that every finite configuration of disjoint simple closed curves in the plane is topologically realizable as the set of limit cycles of a polynomial lienard equation the related vector field x is morse smale moreover it has the minimum number of singularities required for realizing the configuration in a lienard equation we provide an explicit upper bound on the degree of x which is lower than the results obtained before obtained in the context of general polynomial vector fields

approximation from shift invariant subspaces of l 2 ℝ d a complete characterization is given of closed shift invariant subspaces of l2 rd which provide a specified approximation order when such a space is principal i e generated by a single function then this characterization is in terms of the fourier transform of the generator as a special case we obtain the classical strang fix conditions but without requiring the generating function to decay at infinity the approximation order of a general closed shift invariant space is shown to be already realized by a specifiable principal subspace

developments in bivariate spline interpolation the aim of this survey is to describe developments in the field of interpolation by bivariate splines we summarize results on the dimension and the approximation order of bivariate spline spaces and describe interpolation methods for these spaces moreover numerical examples are given

on the stability of normalized powell sabin b splines in this paper we show that the normalized powell sabin b splines form a stable basis for the max norm the approximation constants depend only on the smallest angle in the underlying triangulation since the b splines refer to the size of the powell sabin triangles we find that small powell sabin triangles yield better approximation constants than big powell sabin triangles next in addition to the max norm we treat the lp norm here the approximation constants depend also on a fraction proper to the triangulation thus the b splines are not stable for the lp norm finally as a special case we consider the b spline bases obtained from powell sabin triangles with minimal area and pay extra attention to the approximation constants for the max norm

approximate marginal densities of nonlinear functions this paper presents an asymptotic approximation for the marginal density of a nonlinear function g 0 that is applicable when the joint density of 0 is dominated by a single mode and the jacobian of g is of full rank near that mode the approximation is based on laplace s method and its asymptotic properties are similar to those of the saddlepoint approximation the approximation is applied to the computation of a marginal posterior density a marginal sampling density and a marginal density based on a multivariate saddlepoint approximation to a joint density

small amplitude limit cycles of certain lienard systems the paper is concerned with the number of limit cycles of systems of the form y f x g x where f and g are polynomials for several classes of such systems the maximum number of limit cycles that can bifurcate out of a critical point under perturbation of the coefficients in f and g is obtained in terms of the degree of f and g

approximation by smooth multivariate splines abstract one of the important properties of univariate splines is that in most senses smooth splines approximate just as well as do piecewise polynomials on the same mesh this report shows this to be untrue in the multivariate setting in particular it details the cost in approximating power one may have to pay for the luxury of a smooth piecewise polynomial approximant in an extreme case piecewise polynomials of total degree r on a rectangular grid with all derivatives of order or rho continuous will fail to approximate certain smooth functions at all as the grid goes to zero unless rho is kept below r 3 2 during the analysis of approximation on a certain regular triangular grid a novel kind of bivariate b spline is introduced this b spline in contrast to the established multivariate b spline derived from a simplex can be made to have all its breaklines in a given regular grid this makes it a prime candidate for use in the construction of smooth multivariate piecewise polynomial approximation and its properties will be explored further

geometric treatments and a common mechanism in finite time singularities for autonomous odes abstract geometric treatments of blow up solutions for autonomous ordinary differential equations and their blow up rates are concerned our approach focuses on the type of invariant sets at infinity via compactifications of phase spaces and dynamics on their center stable manifolds in particular we show that dynamics on center stable manifolds of invariant sets at infinity with appropriate time scale desingularizations as well as blowing up of singularities characterize dynamics of blow up solutions as well as their rigorous blow up rates similarities for characterizing finite time extinction and asymptotic behavior of compacton traveling waves to blow up solutions are also shown

a note on posterior moments for a normal mean with double exponential prior summary general results on the behaviour of the posterior mean and variance of the mean of a normal distribution with known variance using a double exponential prior are derived and complement the conjectures and particular examples given by pericchi and smith 1992 and pericchi and walley 1991

a moment of indecision we consider inference from a single observation x about a location parameter 0 that is given 0 0 the density function of x has the form f x 0 for some known function f the prior distribution for 0 has density g 6 it is well known that if f and g are both normal densities then the posterior density is normal with a variance independent of x and less than the variance of either f or g very different behaviour can occur if either component is not normal dawid 1973 studied the limiting behaviour of the posterior distribution as the observation x tends to infinity he showed that for certain forms of f and g depending on the thickness of their tails one component of the posterior will in the limit completely dominate the other then the limiting posterior density is either the prior density g 6 or the likelihood f x 0 in particular he showed that if one of these components is a normal density and the other a t density then the normal component will dominate o hagan 1979 proved that the normal density cannot be dominated by any other dawid s work showed that with a thick tailed f or g the limiting behaviour of the posterior distribution given an extreme observation could be very different from the case when both f and g are normal in the present note we consider the approach to that limit and prove that under certain circumstances the posterior variance undergoes a sharp peak before settling down to its limiting value we interpret this phenomenon intuitively as a manifestation of indecision the observation is large enough for the prior and the data to conflict i e the marginal probability of x being in a neighbourhood of its observed value is very small but not large enough for that conflict to be clearly resolved by one source of information dominating the other box 1980 uses such conflict between prior and data as a means for criticizing the model but we will suppose that the model is not questioned

numerical validation of blow up solutions with quasi homogeneous compactifications we provide a numerical validation method of blow up solutions for finite dimensional vector fields admitting asymptotic quasi homogeneity at infinity our methodology is based on quasi homogeneous compactifications containing quasi parabolic type and directional type compactifications divergent solutions including blow up solutions then correspond to global trajectories of associated vector fields with appropriate time variable transformation tending to equilibria on invariant manifolds representing infinity we combine standard methodology of rigorous numerical integration of differential equations with lyapunov function validations around equilibria corresponding to divergent directions which yields rigorous upper and lower bounds of blow up time as well as rigorous profile enclosures of blow up solutions

topological equivalence of a plane vector field with its principal part defined through newton polyhedra abstract we introduce a blowing up of singularities of vector fields associated with newton polyhedra in the space of the exponents by means of which we prove a generalization of the hartman grobman theorem namely a plane vector field possessing characteristic orbits is locally topologically equivalent with its principal part under suitable non degeneracy hypotheses under stronger hypotheses a similar equivalence result is proven between a plane vector field and a single quasihomogeneous component of its principal part the case of second order equations is also studied

a new criterion for controlling the number of limit cycles of some generalized lienard equations abstract we consider a class of planar differential equations which include the lienard differential equations by applying the bendixson dulac criterion for l connected sets we reduce the study of the number of limit cycles for such equations to the condition that a certain function of just one variable does not change sign as an application this method is used to give a sharp upper bound for the number of limit cycles of some lienard differential equations in particular we present a polynomial lienard system with exactly three limit cycles

quasi interpolation by c1 quartic splines on type 1 triangulations abstract in this paper we construct two new families of c 1 quartic quasi interpolating splines on type 1 triangulations approximating regularly distributed data the splines are directly determined by setting their bernstein bezier coefficients to appropriate combinations of the given data values instead of defining the approximating splines as linear combinations of compactly supported bivariate spanning functions and do not use prescribed derivatives at any point of the domain the quasi interpolation operators provided by the proposed schemes interpolate the data values at the vertices of the triangulation reproduce cubic polynomials and yield approximation order four for smooth functions we also propose some numerical tests that confirm the theoretical results

fully exponential laplace approximations to expectations and variances of nonpositive functions abstract tierney and kadane 1986 presented a simple second order approximation for posterior expectations of positive functions they used laplace s method for asymptotic evaluation of integrals in which the integrand is written as f exp nh and the function h is approximated by a quadratic the form in which they applied laplace s method however was fully exponential the integrand was written instead as exp nh log f this allowed first order approximations to be used in the numerator and denominator of a ratio of integrals to produce a second order expansion for the ratio other second order expansions hartigan 1965 johnson 1970 lindley 1961 1980 mosteller and wallace 1964 require computation of more derivatives of the log likelihood function in this article we extend the fully exponential method to apply to expectations and variances of nonpositive functions to obtain a second order approximation to an expectation e g we use the fully exponential method to approximate

exact and approximate posterior moments for a normal location parameter the forms of first and second posterior moments for a normal location parameter are identified for a rather general class of prior distributions exact and approximate illustrations are given where the prior distribution is double exponential or student t respectively

the period function of classical lienard equations in this paper we study the number of critical points that the period function of a center of a classical lienard equation can have centers of classical lienard equations are related to scalar differential equations x x f x x 0 with f an odd polynomial let us say of degree 2 1 we show that the existence of a finite upperbound on the number of critical periods only depending on the value of can be reduced to the study of slow fast lienard equations close to their limiting layer equations we show that near the central system of degree 2 1 the number of critical periods is at most 2 2 we show the occurrence of slow fast lienard systems exhibiting 2 2 critical periods elucidating a qualitative process behind the occurrence of critical periods it all provides evidence for conjecturing that 2 2 is a sharp upperbound on the number of critical periods we also show that the number of critical periods multiplicity taken into account is always even

normal forms near a symmetric planar saddle connection abstract the present paper studies vector fields of the form x q 2 o 1 x 2 1 x 2 o y y p x o 1 x 2 y o y 2 which contain a separatrix connection between hyperbolic saddles with opposite eigenvalues where the connection is fixed smooth semi local normal forms are provided in vicinity of the connection both in the resonant and non resonant case first a formal conjugacy is constructed near the separatrix then a smooth change of coordinates is realized by generalizing known local results near the hyperbolic points

approximation order from certain spaces of smooth bivariate splines on a three direction mesh determination de l ordre exact d approximation de certains espaces de fonctions splines regulieres a deux variables

classical lienard equations of degree n 6 can have n 12 2 limit cycles abstract based on geometric singular perturbation theory we prove the existence of classical lienard equations of degree 6 having 4 limit cycles it implies the existence of classical lienard equations of degree n 6 having at least n 1 2 2 limit cycles this contradicts the conjecture from lins de melo and pugh formulated in 1976 where an upperbound of n 1 2 limit cycles was predicted this paper improves the counterexample from dumortier panazzolo and roussarie 2007 by supplying one additional limit cycle from degree 7 on and by finding a counterexample of degree 6 we also give a precise system of degree 6 for which we provide strong numerical evidence that it has at least 3 limit cycles

laplace expansion for posterior densities of nonlinear functions of parameters summary an asymptotic expansion for marginal posterior densities of nonlinear functions of parameters is derived the density in question is proportional to an integral over a lower dimensional manifold with the integrand including an appropriate change in volume term laplace s method is applied to obtain the expansion avoiding the difficult step of the explicit parameterization of the manifold we carry out the expansion via an implicit and local parameterization and the derivatives required can be evaluated recursively an example indicates a considerable improvement by including the term of order o n 1

macro elements and stable local bases for splines on clough tocher triangulations macro elements of arbitrary smoothness are constructed on clough tocher triangle splits these elements can be used for solving boundary value problems or for interpolation of hermite data and are shown to be optimal with respect to spline degree we conjecture they are also optimal with respect to the number of degrees of freedom the construction provides local bases for certain superspline spaces defined over clough tocher refinements of arbitrary triangulations these bases are shown to be stable as a function of the smallest angle in the triangulation which in turn implies that the associated spline spaces have optimal order approximation power

on scale mixtures of normal distributions summary the exponential power family of distributions of box tiao 1973 is shown to be a subset of the class of scale mixtures of normals the corresponding mixing distributions are explicitly obtained identifying a close relationship between the exponential power family and a further class of normal scale mixtures namely the stable distributions

accurate approximations for posterior moments and marginal densities abstract this article describes approximations to the posterior means and variances of positive functions of a real or vector valued parameter and to the marginal posterior densities of arbitrary i e not necessarily positive parameters these approximations can also be used to compute approximate predictive densities to apply the proposed method one only needs to be able to maximize slightly modified likelihood functions and to evaluate the observed information at the maxima nevertheless the resulting approximations are generally as accurate and in some cases more accurate than approximations based on third order expansions of the likelihood and requiring the evaluation of third derivatives the approximate marginal posterior densities behave very much like saddle point approximations for sampling distributions the principal regularity condition required is that the likelihood times prior be unimodal

posterior cumulant relationships in bayesian inference involving the exponential family abstract for bayesian inference in one parameter contexts where either the likelihood or the prior has an exponential family form relationships are derived for posterior moments and cumulants of functions of both the canonical and the expectation parameters the identities exhibited generalize the simple relationships well known in the conjugate analysis case applications of these results are indicated in the areas of bayesian robustness and approximation in particular results are obtained on the behavior of the posterior distribution for a large observation generalizing work of meeden and isaacson

robust sequential approximate bayesian estimation summary an approximation to the sequential updating of the distribution of location parameters of a linear time series model is developed for non normal observations the behaviour of the resulting non linear recursive filtering algorithm is examined and shown to have certain desirable properties for a variety of non normal error distributions illustrative examples are given and relationships with previous work on robustness and sequential estimation are mentioned we consider here the problem of sequential estimation of the location vector of a linear time series model termed the dynamic linear model by harrison and stevens 1976 the straightforward exact analysis obtained by assuming normal error and prior structure is generally lost when alternative error distributions are adopted and yet considerations of realism or robustness may strongly suggest such non normal assumptions the multi state model of harrison and stevens provides an approximate analysis based on a discrete variance mixture of normal distributions an approach which has been extensively investigated in the engineering literature under the name of gaussian sum approximations see for example alspach and sorenson 1971 our aim in this paper is to provide an approximate tractible recursive updating procedure for the location parameters which is applicable to a wide range of symmetric unimodal error distributions and is both more realistic and more elegant than the discrete gaussian sum approach in particular for heavy tailed distributions our procedures provide approximate bayesian methods for time series analysis which extend considerably the work of masreliez and martin 1977 and have close connections with classical robustness ideas such as m estimation and influence functions

hilbert s 16th problem for classical lienard equations of even degree abstract classical lienard equations are two dimensional vector fields on the phase plane or on the lienard plane related to scalar differential equations x f x x x 0 in this paper we consider f to be a polynomial of degree 2 l 1 with l a fixed but arbitrary natural number the related lienard equation is of degree 2l we prove that the number of limit cycles of such an equation is uniformly bounded if we restrict f to some compact set of polynomials of degree exactly 2 l 1 the main problem consists in studying the large amplitude limit cycles of which we show that there are at most l

bivariate box splines and smooth pp functions on a three direction mesh abstract let s denote the space of bivariate piecewise polynomial functions of degree k and smoothness on the regular mesh generated by the three directions 1 0 0 1 1 1 we construct a basis for s in terms of box splines and truncated powers this allows us to determine the polynomials which are locally contained in s and to give upper and lower bounds for the degree of approximation for 2k 2 3 k 2 3 the case of minimal degree k for given smoothness we identify the elements of minimal support in s and give a basis for s loc f s supp f with a convex subset of r 2