pseudo normal form near saddle center or saddle focus equilibria abstract in this paper we introduce the pseudo normal form which generalizes the notion of normal form around an equilibrium its convergence is proved for a general analytic system in a neighborhood of a saddle center or a saddle focus equilibrium point if the system is hamiltonian or reversible this pseudo normal form coincides with the birkhoff normal form so we present a new proof in these celebrated cases from the convergence of the pseudo normal form for a general analytic system several dynamical consequences are derived like the existence of local invariant objects

a variable selection heuristic for k means clustering one of the most vexing problems in cluster analysis is the selection and or weighting of variables in order to include those that truly define cluster structure while eliminating those that might mask such structure this paper presents a variable selection heuristic for nonhierarchical k means cluster analysis based on the adjusted rand index for measuring cluster recovery the heuristic was subjected to monte carlo testing across more than 2200 datasets with known cluster structure the results indicate the heuristic is extremely effective at eliminating masking variables a cluster analysis of real world financial services data revealed that using the variable selection heuristic prior to the k means algorithm resulted in greater cluster stability

irregular dynamics and homoclinic orbits to hamiltonian saddle centers we consider 4 dimensional real analytic hamiltonian systems with a saddle center equilibrium related to a pair of real and a pair of imaginary eigenvalues and a homoclinic orbit to it we find conditions for the existence of transversal homoclinic orbits to periodic orbits of long period in every energy level sufficiently close to the energy level of the saddle center equilibrium we also consider one parameter families of reversible 4 dimensional hamiltonian systems we prove that the set of parameter values where the system has homoclinic orbits to a saddle center equilibrium has no isolated points we also present similar results for systems with heteroclinic orbits to saddle center equilibria 1997 john wiley sons inc

detection of symmetric homoclinic orbits to saddle centres in reversible systems abstract we present a perturbation technique for the detection of symmetric homoclinic orbits to saddle centre equilibria in reversible systems of ordinary differential equations we assume that the unperturbed system has primary symmetric homoclinic orbits which may be either isolated or appear in a family and use an idea similar to that of melnikov s method to detect homoclinic orbits in their neighbourhood this technique also allows us to identify bifurcations of unperturbed or perturbed symmetric homoclinic orbits our technique is of importance in applications such as nonlinear optics and water waves since homoclinic orbits to saddle centre equilibria describe embedded solitons ess in systems of partial differential equations representing physical models and except for special cases their existence has been previously studied only numerically using shooting methods and continuation techniques we apply the general theory to two examples a four dimensional system describing ess in nonlinear optical media and a six dimensional system which can possess a one parameter family of symmetric homoclinic orbits in the unperturbed case for these examples the analysis is compared with numerical computations and an excellent agreement between both results is found

a c r trivariate macro element based on the worsey farin split of a tetrahedron we construct trivariate c r macro elements for any r ge 1 over the worsey farin refinement of any tetrahedral partition this extends the construction of c 1 cubic worsey farin elements and c 2 elements of degree nine to the c r situation with r 2 we also show that the degree of polynomials used for our macro elements is the lowest

existence of exponentially small separatrix splittings and homoclinic connections between whiskered tori in weakly hyperbolic near integrable hamiltonian systems abstract in this paper we study a generalization of arnold s original example in which he discussed the existence of a mechanism for instability caused by the splitting of the homoclinic manifolds of the weakly hyperbolic tori that has subsequently been referred to as arnold diffusion in case when the number of degrees of freedom n 3 namely we consider a widely studied model of a pendulum weakly coupled with n 1 rotors with the degeneracies in the unperturbed hamiltonian corresponding to different timescales existing in the problem using an alloy of the iterative and direct methods developed within the last years we give exponentially small upper bounds for the splitting measure of transversality for the case of an even analytic perturbation thus improving the estimate of gallavotti 1994 which he calls quasiflat and generalizing the analogous recent estimate of delshams et al 1996 for the rapidly quasiperiodically forced pendulum to a much larger class of hamiltonian systems in particular the exponentially small upper bound for the transversality measure of the splitting applies when the hamiltonian has extra degeneracies namely when the frequencies on a torus become near resonant in fact we show that in such a case the quantity in question becomes smaller which is the incarnation of the general fact that resonant regions in the action space are more stable in the sense that they have larger nekhoroshev exponent nekhoroshev exponent nevertheless we emphasize that getting uniform estimates for an arbitrary n 3 is very hard unless one makes some additional assumptions on the approximation properties of the frequency vector although recent developments show that the first order of canonical perturbation theory given by melnikov integrals generally cannot be accepted as the leading order answer for the splitting distance for the case of more than two degrees of freedom including the rapidly quasiperiodically forced pendulum problem we suggest an analytic perturbation the majority of whose fourier components are strictly nondashzero for which melnikov integrals can be vindicated as the leading order approximation for the components of the splitting distance in different directions if the frequencies on the invariant tori satisfy certain arithmetic conditions this allows us to bound the splitting distance from below furthermore having such a perturbation for the case of three degrees of freedom we use a simple number theoretical argument to find the asymptotics of the fourier series with exponentially small coefficients involved this enables us to compute the numerous homoclinic orbits for the whiskered tori of asymptotically full measure and by proving the domineering contribution of the first order of perturbation theory for the transversality measure to suggest a leading order answer for this quantity which implies the existence of an infinite number of heteroclinic connections between tori with close diophantine frequencies we elucidate the numerous arithmetic issues that obstruct getting a compact leading order approximation for the splitting size most of which can be overcome in the case of three degrees of freedom as our example shows these obstacles can be also possibly avoided in the same fashion for an arbitrary n 3 if one treats the case when the frequencies of the rotors are near a resonance of multiplicity n 3 or n 2

local analytic reduction of families of diffeomorphisms abstract we study local analytic simplification of families of analytic maps near a hyperbolic fixed point a particularly important application of the main result concerns families of hyperbolic saddles where siegel s theorem is too fragile at least in the analytic category by relaxing on the formal normal form we obtain analytic conjugacies since we consider families it is more convenient to state some results for analytic maps on a banach space this gives no extra complications as an example we treat a family passing through a 1 1 resonant saddle

nonintegrability of some hamiltonian systems scattering and analytic continuation we consider canonical two degrees of freedom analytic hamiltonian systems with hamiltonian functionh 1 2 p12 p22 u q1 q2 where u q1 q2 1 2 v2q12 2q22 o q12 q22 3 2 and q2 u q1 0 0 under some additional not so restrictive hypothesis we present explicit conditions for the exisstence of transversal homoclinic orbits to some periodic orbits of these systems we use a theorem of lerman 1991 and an analogy between one of its conditions with the usual one dimensional quantum scattering problem the study of the scattering equation leads us to an analytic continuation problem for the solutions of a linear second order differential equation we apply our results to some classical problems

horseshoes in two degree of freedom hamiltonian systems with saddle centers we consider a class of two degree of freedom hamiltonian systems which have saddle centers with homoclinic orbits and do not take the form of small perturbations of integrable systems using a melnikov type global perturbation technique we present a condition under which orbits transversely homoclinic to periodic orbits exist and horseshoes are created on surfaces of energy level near the saddle centers we apply the theory to systems with potentials and give concrete examples in particular we can detect the presence of horseshoes in a class of systems to which previous results did not apply

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

exponentially small splitting of separatrices under fast quasiperiodic forcing we consider fast quasiperiodic perturbations with two frequencies 1 epsiv of a pendulum where is the golden mean number the complete system has a two dimensional invariant torus in a neighbourhood of the saddle point we study the splitting of the three dimensional invariant manifolds associated to this torus provided that the perturbation amplitude is small enough with respect to and some of its fourier coefficients the ones associated to fibonacci numbers are separated from zero it is proved that the invariant manifolds split and that the value of the splitting which turns out to be exponentially small with respect to is correctly predicted by the melnikov function

smooth macro elements on powell sabin 12 splits macro elements of smoothness c r are constructed on powell sabin 12 splits of a triangle for all r 0 these new elements complement those recently constructed on powell sabin 6 splits and can be used to construct convenient superspline spaces with stable local bases and full approximation power that can be applied to the solution of boundary value problems and for interpolation of hermite data

dynamics around simple resonant tori in nearly integrable hamiltonian systems abstract by an application of the k a m theory we derive an accurate normal form valid in the vicinity of partially hyperbolic tori which arise close to simple resonances in nearly integrable hamiltonian systems this normal form allows to detect orbits homoclinic to a persistent torus moreover it also gives precise estimates on the times of transition around the stable and unstable manifolds of these tori hence we provide an efficient tool to compute the speed of drift of orbits shadowing a chain of hyperbolic tori associated to a simple resonant curve in the action space

composite finite elements of class ck let be a triangulation of d r2 with vertices a ai 1 i n and pkn d v ck d t v t pn where pn polynomials of total degree n for u cr d r k and v finite dimensional subspace of ck d we consider the hermite problem hk a v find v v d v ai d u ai 1 i n k this problem is solvable in v pkn d iff n 4k 1 enisek when v pkn d 3 where 3 is a subtriangulation of obtained by subdividing each t into 3 triangles the problem has a solution for n 4k 1 e g the c1 cubic hct triangle and a p27 triangle with 31 parameters when v pkn d 6 where 6 is a subtriangulation of obtained by subdividing each t into 6 triangles the problem has a solution for n 3k 1 e g the c1 quadratic ps triangle and a p25 triangle with 31 parameters in both cases the construction needs the partial derivatives d u ai for 2k 1 the domain d is assumed to be polygonal compact and connected

a c 2 trivariate macroelement based on the worsey farin split of a tetrahedron a c 2 trivariate macroelement is constructed based on the worsey farin split of a tetrahedron into twelve subtetrahedra the element uses supersplines of degree 9 and provides optimal order approximation of smooth functions

exponentially small splitting of separatrices beyond melnikov analysis rigorous results in this paper we study the problem of exponentially small splitting of separatrices of one degree of freedom classical hamiltonian systems with a non autonomous perturbation which is fast and periodic in time we provide a result valid for general systems which are polynomials or trigonometric polynomials in the state variables our result consists in obtaining a rigorous proof of the asymptotic formula for the measure of the splitting we have obtained that the splitting has the asymptotic behavior k e a identifying the constants k and a in terms of the features of the system the study of our problem leads us to consider several cases in some cases assuming the per turbation is small enough it turns out that the values of k coincides with the classical melnikov approach we have identified the limit size of the perturbation for which this classical theory holds true however for the limit cases which appear naturally both in averaging theory and bifurcation theory we encounter that generically neither k nor are actually well predicted by melnikov theory

smooth macro elements based on clough tocher triangle splits summary macro elements of smoothness c r on clough tocher triangle splits are constructed for all r ge 0 these new elements are improvements on elements constructed in 11 in that disproving a conjecture made there certain unneeded degrees of freedom have been removed numerical experiments on hermite interpolation with the new elements are included

separatrix splitting for systems with three time scales an exact expression for the determinant of the splitting matrix is derived for three degrees of freedom systems with three time scales it allows us to analyze the asymptotic behaviour needed to amend the large angles theorem proposed in ann inst h poincare b 60 1 1994 the asymptotic validity of mel nikov s integrals is proved for the class of models considered which are polynomial perturbations the technique for exhibiting cancellations is inspired by renormalization theory in quantum electrodynamics and uses an analogue of dyson s equations to prove an infinite family of identities due to symmetries that remind us of ward s identities

on the reducibility of linear differential equations with quasiperiodic coefficients the system x a eq t x in rd is considered where a is a constant matrix and q a quasiperiodic analytic matrix with r basic frequencies the eigenvalues of a are arbitrary including the purely imaginary case suppose that the set formed by the eigenvalues of a and the basic frequencies of q satisfies a nonresonant condition then there is a positive measure cantorian set e such that for e e the system is reducible to constant coefficients by means of a quasiperiodic change of variables provided a nondegeneracy condition holds this condition prevents locking at resonance

arnold diffusion in a priori chaotic symplectic maps we assume that a symplectic real analytic map has an invariant normally hyperbolic cylinder and an associated transverse homoclinic cylinder we prove that generically in the real analytic category the boundaries of the invariant cylinder are connected by trajectories of the map

gaussian quadrature for c1 cubic clough tocher macro triangles abstract a numerical integration rule for multivariate cubic polynomials over n dimensional simplices was designed by hammer and stroud 1956 the quadrature rule requires n 2 quadrature points the barycentre of the simplex and n 1 points that lie on the connecting lines between the barycentre and the vertices of the simplex in the planar case this particular rule belongs to a two parameter family of quadrature rules that admit exact integration of bivariate polynomials of total degree three over triangles we prove that this rule is exact for a larger space namely the c 1 cubic clough tocher spline space over macro triangles if and only if the split point is the barycentre this results in a factor of three reduction in the number of quadrature points needed to integrate the clough tocher spline space exactly

rapidly forced planar vector fields and splitting of separatrices abstract we prove that for analytic rapidly forced two dimensional systems the melnikov function is exponentially small with respect to the period we provide an example of non validity of melnikov s method and we obtain upper bounds for conservative and analytic planar systems for big amplitude periodic rapidly forcing

on homoclinic splitting problems abstract counterexamples to the homoclinic splitting claimed in the paper existence of exponentially small separatrix settings and homoclinic connections between whiskered tori in weakly hyperbolic near integrable hamiltonian systems by rudnev and wiggins physica d 114 1998 3 80 are given these counterexamples invalidate the proofs of the main results of that paper

on the conservation of adiabatic invariants for a system of coupled rotators abstract we study the accuracy of the conservation of adiabatic invariants in a model of n weakly coupled rotators most attention is devoted to n 2 and frequency 1 2 with 2 1 quadratic irrational we apply a heuristic approximation scheme going back to jeans and to landau and teller and perform a very accurate numerical check of the result observing a quite remarkable agreement

a proof of the exponentially small transversality of the separatrices for the standard map in 1984 v f lazutkin laz84 lst89 obtained an asymptotic formula for the separatrix splitting angle for the standard map the difficulty of this problem is related to the exponential smallness of the splitting with respect to a perturbation parameter lazutkin s proof was based on two conjectures probably the original form of those conjectures was incorrect but lazutkin s method was very efficient and inspired a large number of studies on the exponentially small splitting of separatrices the consequent works laz91 laz92 gls94 and gel96 prepared the base to fill all the gaps of the original proof the present paper contains a complete and self contained proof of a refined version of the original formula formula 1 7 of the present paper in this form the formula was obtained in gls94 the proof is inspired by the ideas of lazutkin s original paper laz84

an examination of procedures for determining the number of clusters in a data set a monte carlo evaluation of 30 procedures for determining the number of clusters was conducted on artificial data sets which contained either 2 3 4 or 5 distinct nonoverlapping clusters to provide a variety of clustering solutions the data sets were analyzed by four hierarchical clustering methods external criterion measures indicated excellent recovery of the true cluster structure by the methods at the correct hierarchy level thus the clustering present in the data was quite strong the simulation results for the stopping rules revealed a wide range in their ability to determine the correct number of clusters in the data several procedures worked fairly well whereas others performed rather poorly thus the latter group of rules would appear to have little validity particularly for data sets containing distinct clusters applied researchers are urged to select one or more of the better criteria however users are cautioned that the performance of some of the criteria may be data dependent

real submanifolds of maximum complex tangent space at a cr singular point i we study a germ of real analytic n dimensional submanifold of mathbf c n that has a complex tangent space of maximal dimension at a cr singularity under some assumptions we show its equivalence to a normal form under a local biholomorphism at the singularity we also show that if a real submanifold is formally equivalent to a quadric it is actually holomorphically equivalent to it if a small divisors condition is satisfied finally we investigate the existence of a complex submanifold of positive dimension in mathbf c n that intersects a real submanifold along two totally and real analytic submanifolds that intersect transversally at a possibly non isolated cr singularity

an asymptotic expression for the splitting of separatrices of the rapidly forced pendulum the measure of the splitting of the separatrices of the rapidly forced pendulum ddot x sin x mu sin frac t varepsilon is considered as a model problem that has been studied by different authors here e are small parameters e 0 but otherwise independent the following formula for the angle between separatrices is established alpha frac pi 2 varepsilon frac mu cosh frac pi 2 varepsilon left 1 o mu varepsilon 2 right

a bivariate c1 cubic super spline space on powell sabin triangulation a bivariate c 1 cubic super spline is constructed on powell sabin type 1 split with the additional smoothness at vertices in the original triangulation being c 2 which permits the hermite interpolation up to the second order partial derivatives exactly on all the vertices in the original triangulation the locally supported dual basis and computational details using derivatives around each vertex are given

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 super splines and finite elements super splines are spaces of multivariate splines consisting of piecewise polynomials defined on triangulations where the degree of smoothness enforced at the vertices is larger than the degree of smoothness enforced across the edges dimension formulae for super spline spaces are established and the construction of explicit bases of locally supported splines is presented in addition it is shown that many classical finite element spaces can be interpreted as super spline spaces providing a link between spline theory and the highly applicable finite element theory

on numerical quadrature for c1 quadratic powell sabin 6 split macro triangles abstract the quadrature rule of hammer and stroud 1956 for cubic polynomials has been shown to be exact for a larger space of functions namely the c 1 cubic clough tocher spline space over a macro triangle if and only if the split point is the barycentre of the macro triangle kosinka and barto 2018 we continue the study of quadrature rules for spline spaces over macro triangles now focusing on the case of c 1 quadratic powell sabin 6 split macro triangles we show that the 3 node gaussian quadrature s for quadratics can be generalised to the c 1 quadratic powell sabin 6 split spline space over a macro triangle for a two parameter family of inner split points not just the barycentre as in kosinka and barto 2018 the choice of the inner split point uniquely determines the positions of the edge split points such that the whole spline space is integrated exactly by a corresponding polynomial quadrature consequently the number of quadrature points needed to exactly integrate this special spline space reduces from twelve to three for the inner split point at the barycentre we prove that the two 3 node quadratic polynomial quadratures of hammer and stroud exactly integrate also the c 1 quadratic powell sabin spline space if and only if the edge split points are at their respective edge midpoints for other positions of the inner and edge split points we provide numerical examples showing that three nodes suffice to integrate the space exactly but a full classification and a closed form solution in the generic case remain elusive

a local lagrange interpolation method based on c 1 cubic splines on freudenthal partitions a trivariate lagrange interpolation method based on c 1 cubic splines is described the splines are defined over a special refinement of the freudenthal partition of a cube partition the interpolating splines are uniquely determined by data values but no derivatives are needed the interpolation method is local and stable provides optimal order approximation and has linear complexity

validating clusters with the lower bound for sum of squares error given that a minor condition holds e g the number of variables is greater than the number of clusters a nontrivial lower bound for the sum of squares error criterion in k means clustering is derived by calculating the lower bound for several different situations a method is developed to determine the adequacy of cluster solution based on the observed sum of squares error as compared to the minimum sum of squares error

cr rational splines over a triangulation abstract we propose a construction of c r bivariate rational splines over a triangulation via a finite element approach for r 2 the usual bivariate polynomial splines require some supersmoothness at the vertices of the triangles even when dealing with macro elements this is no more a requirement for our rational approach in this paper we give a constructive method for rational c r finite elements in r 2 defined over triangles beside the general case the process of construction is illustrated with two examples namely the c 1 case and the c 2 case

an algorithm for generating artificial test clusters an algorithm for generating artificial data sets which contain distinct nonoverlapping clusters is presented the algorithm is useful for generating test data sets for monte carlo validation research conducted on clustering methods or statistics the algorithm generates data sets which contain either 1 2 3 4 or 5 clusters by default the data are embedded in either a 4 6 or 8 dimensional space three different patterns for assigning the points to the clusters are provided one pattern assigns the points equally to the clusters while the remaining two schemes produce clusters of unequal sizes finally a number of methods for introducing error in the data have been incorporated in the algorithm

a practical use of the melnikov homoclinic method using cutoff functions and periodic extensions we prove that the melnikov homoclinic method gives a criterium to show that for a finite time interval t t with t arbitrarily large the perturbed system is conjugated to a chaotic one for quite general classes of perturbation functions the method is applied to specific perturbations of the pendulum and of the gylden s problem

total convergence or general divergence in small divisors we study generic holomorphic families of dynamical systems presenting problems of small divisors with fixed arithmetic the characteristic features are delicate problems of convergence of formal power series due to small divisors we prove the following dichotomy we have convergence for all parameter values or divergence everywhere except for an exceptional pluri polar set of parameters we illustrate this general principle in different problems of small divisors as an application we obtain new richer families of non linearizable examples in the siegel problem when the bruno condition is violated generalizing and extending to higher dimension previous results of yoccoz and the author

estimating the number of clusters in a data set via the gap statistic we propose a method the gap statistic for estimating the number of clusters groups in a set of data the technique uses the output of any clustering algorithm e g k means or hierarchical comparing the change in within cluster dispersion with that expected under an appropriate reference null distribution some theory is developed for the proposal and a simulation study shows that the gap statistic usually outperforms other methods that have been proposed in the literature

on statistical data analysis based on the l1 norm and related methods an introduction to statistical data analysis l 1 norm based y dodge the place of the l 1 norm in robust estimation p j huber i historical development the historical development of the l 1 and l estimation procedures r w farebrother bounded influence inference in regression a review e ronchetti ii computational algorithms and computer packages algorithms for unconstrained l 1 linear regression j e gentle s c narula v a sposito the reduced gradient algorithm m r osborne the l 1 estimate as limiting case of an l p or huber estimate h ekblom a review of computational methods for solving the nonlinear l 1 norm estimation problem r gonin a h money fitting data through homotopy methods j p schellhorn blinwdr a fortran program for robust and bounded influence regression r dutter solving bounded influence regression problems with robsys a marazzi xplore a computing environment for exploratory regression w hardle iii estimation characterization properties and selection of variables l 1 distances in statistical inference comparison of topological functional and statistical properties i vajda the role played by l 1 in data analysis b fichet l 1 embeddings of a data structure i d g le calve assessing the accuracy of the sample median estimated standard errors vs interpolated confidence intervals s j sheather the median of a finite measure on a banach space j h b kemperman variable selection in linear model based on trimmed least squares estimator j antoch numerical techniques for finding estimates which minimize the upper bound of the absolute deviation a gaivoronski asymptotic properties of restricted l 1 estimates of regression j dupacova adaptive combination of least squares and least absolute deviations estimators y dodge j jureckova iv statistical inference procedures a comparison of asymptotic testing methods for l 1 regression r koenker least absolute errors analysis of variance j w mckean r m schrader small sample properties of least absolute errors analysis of variance r m schrader j w mckean bootstrap and inference procedures for l 1 regression g stangenhaus invariant tests in bivariate models and the l 1 criterion b m brown t p hettmansperger using regression fractiles to identify outliers s portnoy various discrepancy measures in model testing two competing regression models v v fedorov v nonparametric analysis kernel estimates of the sparsity function a h welsh what does optimal bandwidth selection mean for nonparametric regression estimation j s marron density estimation from dependent sample l gyorfi vi cluster analysis clustering by means of medoids l kaufman p j rousseeuw l 1 in fuzzy clustering e trauwaert using the l 1 norm within cluster analysis h spath vii applications an application of l 1 to astronomy p j rousseeuw

c1 quintic interpolation over triangles two explicit representations two explicit representation of a c1 quintic interpolant over triangles are given these representations are derived by generalization of coons methods and bernstein bezier methods respectively