saari s homographic conjecture of the three body problem saari s homographic conjecture which extends a classical statement proposed by donald saari in 1970 claims that solutions of the newtonian n body problem with constant configurational measure are homographic in other words if the mutual distances satisfy a certain relationship the configuration of the particle system may change size and position but not shape we prove this conjecture for large sets of initial conditions in three body problems given by homogeneous potentials including the newtonian one some of our results are true for n 3

w stability and local input output stability results a new type of input output stability is defined based on the use of a sobolev space w w is well suited like the lebesgue space l sub 2 to obtain stability characterizations in the time and frequency domains moreover if compared with l sub 2 w has additional properties which enable one to establish local stability results a local version of the small gain theorem is established in this framework as well as some consequences of this result in particular local versions of the passivity theorem and of the circle criterion the relationship between w stability and asymptotic stability is studied

index and stability of symmetric periodic orbits in hamiltonian systems with application to figure eight orbit in this paper using the maslov index theory in symplectic geometry we build up some stability criteria for symmetric periodic orbits in a hamiltonian system which is motivated by the recent discoveries in the n body problem the key ingredient is a generalized bott type iteration formula for periodic solution in the presence of finite group action on the orbit for second order system we prove under general boundary conditions the close formula for the relationship between the morse index of an orbit in a lagrangian system and the maslov index of the fundamental solution for the corresponding orbit in its hamiltonian system counterpart and the boundary conditions cover the cases which appeared in the n body problem as an application we consider the stability problem of the celebrated figure eight orbit due to chenciner and montgomery in the planar three body problem with equal masses and we clarify the relationship between linear stability and its variational nature on various loop spaces the basic idea is as follows the variational characterization of the figure eight orbit provides information about its morse index based on its relation to the maslov index our stability criteria come into play

on the computation of sets of points with low lebesgue constant on the unit disk abstract in this paper of numerical nature we test the lebesgue constant of several available point sets on the disk and propose new ones that enjoy low lebesgue constant furthermore we extend some results in cuyt 2012 analyzing the case of bos arrays whose radii are nonnegative gauss gegenbauer lobatto nodes with exponent noticing that the optimal still allows to achieve point sets on with low lebesgue constant n for degrees n 30 next we introduce an algorithm that through optimization determines point sets with the best known lebesgue constants for n 25 finally we determine theoretically a point set with the best lebesgue constant for the case n 1

further facts about input to state stabilization previously published results about input to state stabilizability are shown to hold even for systems which are not linear in controls provided that a more general type of feedback is allowed applications to certain stabilization problems and coprime factorizations as well as comparisons to other results on input to state stability are briefly discussed

constructing least squares polynomial approximations polynomial approximations constructed using a least squares approach form a ubiquitous technique in numerical computation one of the simplest ways to generate data for least squares problems is wi

star pentagon and many stable choreographic solutions of the newtonian 4 body problem abstract in this paper we give a rigorous proof of the existence of infinitely many simple choreographic solutions in the classical newtonian 4 body problem these orbits are discovered by a variational method with structural prescribed boundary conditions spbc this method provides an initial path that is obtained by minimizing the lagrangian action functional over the spbc we prove that the initial path can be extended to a periodic or quasi periodic solution with computer assistance a family of choreographic orbits of this type is shown to be linearly stable among the many linearly stable simple choreographic orbits the most extraordinary one is the stable star pentagon choreographic solution we also prove the existence of infinitely many double choreographic periodic solutions infinitely many non choreographic periodic solutions and uncountably many quasi periodic solutions each type of periodic solutions has many stable solutions and possibly infinitely many stable solutions our results with spbc largely complement the current results by minimizing the action on a loop space

saari s conjecture for the collinear n body problem in 1970 don saari conjectured that the only solutions of the newtonian n body problem that have constant moment of inertia are the relative equilibria we prove this conjecture in the collinear case for any potential that involves only the mutual distances furthermore in the case of homogeneous potentials we show that the only collinear and non zero angular momentum solutions are homographic motions with central configurations

new relationships between input output and lyapunov stability in this note we define a concept called small signal l p stability and show its relationship to lyapunov stability

sobolev norms based state estimation and input recovery for a class of nonlinear systems design and experimental results in this paper we address the problem of state estimation and input recovery for a class of nonlinear systems in the presence of disturbances in both the state and output equations indeed one of the main difficulties that arise when input recovery is considered is how to cope with the problem of disturbance s derivative for the filter design our contribution lies in the use of sobolev norms to develop a simple and useful observer we provide first the state filtering and input recovery equations after based on the lyapunov stability theory and some robustness criteria new sufficient synthesis conditions are given in terms of linear matrix inequalities lmis to show performances of the proposed method we considered the problem of simultaneous synchronization and decryption in chaotic communication systems with some experimental results

on the generation of symmetric lebesgue like points in the triangle we compute point sets on the triangle that have low lebesgue constant with sixfold symmetries and gauss legendre lobatto distribution on the sides up to interpolation degree 18 such points have the best lebesgue constants among the families of symmetric points used so far in the framework of triangular spectral elements

non planar minimizers and rotational symmetry in the n body problem abstract the newtonian n body problem admits uniformly rotating relative equilibrium solutions in the plane but not in r 3 when the bodies are allowed to interact in r 3 is it preferable in terms of action to leave the plane and follow a non planar trajectory we use the variational techniques of chenciner and venturelli celestial mech dyn astro 77 2000 139 to show that for an open set of masses there is a class of collision free action minimizing orbits of certain rotational symmetry in the four body problem which are non coplanar i e the planar relative equilibrium is not the least action solution among orbits in r 3 both periodic and quasi periodic solutions are constructed in this way we also discuss constructing collision free action minimizing solutions possessing d rotational symmetry along with various other symmetry constraints

a multiloop generalization of the circle criterion for stability margin analysis in order to provide a theoretical tool well suited for use in characterizing the stability margins e g gain and phase margins of multiloop feedback systems multiloop input output stability results generalizing the circle stability criterion are considered generalized conic sectors with centers and radii determined by linear dynamical operators are employed to enable an engineer to specify the stability margins which he desires as a frequency dependent convex set of modeling errors including nonlinearities gain variations and phase variations which the system must be able to tolerate in each feedback loop without instability the resulting stability criterion gives sufficient conditions for closed loop stability in the presence of such frequency dependent modeling errors even when the modeling errors occur simultaneously in all loops so for example stability is assured as loop gains and phases vary throughout a set of nonzero measure whose boundaries are frequency dependent the stability conditions yield an easily interpreted scalar measure of the amount by which a muitiloop system exceeds or falls short of its stability margin specifications

a remarkable periodic solution of the three body problem in the case of equal masses using a variational method we exhibit a surprisingly simple periodic orbit for the newtonian problem of three equal masses in the plane the orbit has zero angular momentum and a very rich symmetry pattern its most surprising feature is that the three bodies chase each other around a flxed eight shaped curve setting aside collinear motions the only other known motion along a flxed curve in the inertial plane is the lagrange relative equilibrium in which the three bodies form a rigid equilateral triangle which rotates at constant angular velocity within its circumscribing circle our orbit visits in turns every euler conflguration in which one of the bodies sits at the midpoint of the segment deflned by the other two figure 1 numerical computations

local l sub p stability and local small gain theorem for discrete time systems the notion of local l sub 2 stability is defined the relationship between this notion and lyapunov stability is clarified a local version of the small gain theorem is then established in the case of discrete time systems these results are applied to stability analysis of a nonlinear discrete time delay system

a cardinal function algorithm for computing multivariate quadrature points we present a new algorithm for numerically computing quadrature formulas for arbitrary domains which exactly integrate a given polynomial space an effective method for constructing quadrature formulas has been to numerically solve a nonlinear set of equations for the quadrature points and their associated weights symmetry conditions are often used to reduce the number of equations and unknowns our algorithm instead relies on the construction of cardinal functions and thus requires that the number of quadrature points n be equal to the dimension of a prescribed lower dimensional polynomial space the cardinal functions allow us to treat the quadrature weights as dependent variables and remove them as well as an equivalent number of equations from the numerical optimization procedure we give results for the triangle where for all degrees d le 25 we find quadrature formulas of this form which have positive weights and contain no points outside the triangle seven of these quadrature formulas improve on previously known results

an algorithm for computing fekete points in the triangle on the line and its tensor products fekete points are known to be the gauss lobatto quadrature points but unlike high order quadrature fekete points generalize to non tensor product domains such as the triangle thus fekete points might serve as an alternative to the gauss lobatto points for certain applications in this work we present a new algorithm to compute fekete points and give results up to degree 19 for the triangle for degree d 10 these points have the smallest lebesgue constant currently known the computations validate a conjecture of bos j approx theory 64 1991 pp 271 280 that fekete points along the boundary of the triangle are the one dimensional gauss lobatto points

polynomial chaos expansions for dependent random variables abstract polynomial chaos expansions pce are well suited to quantifying uncertainty in models parameterized by independent random variables the assumption of independence leads to simple strategies for building multivariate orthonormal bases and for sampling strategies to evaluate pce coefficients in contrast the application of pce to models of dependent variables is much more challenging three approaches can be used to construct pce of models of dependent variables the first approach uses mapping methods where measure transformations such as the nataf and rosenblatt transformation can be used to map dependent random variables to independent ones however we show that this can significantly degrade performance since the jacobian of the map must be approximated a second strategy is the class of dominating support methods in these approaches a pce is built using independent random variables whose distributional support dominates the support of the true dependent joint density we provide evidence that this approach appears to produce approximations with suboptimal accuracy a third approach the novel method proposed here uses gram schmidt orthogonalization gso to numerically compute orthonormal polynomials for the dependent random variables this approach has been used successfully when solving differential equations using the intrusive stochastic galerkin method and in this paper we use gso to build pce using a non intrusive stochastic collocation method the stochastic collocation method treats the model as a black box and builds approximations of the input output map from a set of samples building pce from samples can introduce ill conditioning which does not plague stochastic galerkin methods to mitigate this ill conditioning we generate weighted leja sequences which are nested sample sets to build accurate polynomial interpolants we show that our proposed approach gso with weighted leja sequences produces pce which are orders of magnitude more accurate than pce constructed using mapping or dominating support methods

a comparison of high order interpolation nodes for the pyramid the use of pyramid elements is crucial to the construction of efficient hex dominant meshes m bergot g cohen and m durufle j sci comput 42 2010 pp 345 381 for conforming nodal finite element methods with mixed element types it is advantageous for nodal distributions on the faces of the pyramid to match those on the faces and edges of hexahedra and tetrahedra we adapt existing procedures for constructing optimized tetrahedral nodal sets for high order interpolation to the pyramid with constrained face nodes including two generalizations of the explicit warp and blend construction of nodes on the tetrahedron t warburton j engrg math 56 2006 pp 247 262 comparisons between nodal sets show that the lowest lebesgue constants are given by warp and blend nodes for order n leq 7 and fekete nodes for n 7 though numerical experiments show little variation in the conditioning and accuracy of all surveyed nonequidistant points

trace formula for linear hamiltonian systems with its applications to elliptic lagrangian solutions in the present paper we build up trace formulas for both the linear hamiltonian systems and sturm liouville systems the formula connects the monodromy matrix of a symmetric periodic orbit with the infinite sum of eigenvalues of the hessian of the action functional a natural application is to study the non degeneracy of linear hamiltonian systems precisely by the trace formula we can give an estimation for the upper bound such that the non degeneracy preserves moreover we could estimate the relative morse index by the trace formula consequently a series of new stability criteria for the symmetric periodic orbits is given as a concrete application the trace formula is used to study the linear stability of elliptic lagrangian solutions of the classical planar three body problem which depends on the mass parameter beta in 0 9 and the eccentricity e in 0 1 based on the trace formula we estimate the stable region and hyperbolic region of the elliptic lagrangian solutions

existence and linear stability of the rhomboidal periodic orbit in the planar equal mass four body problem abstract in this paper we study the existence and linear stability of the rhomboidal periodic orbit in the planar equal mass four body problem the hamiltonian of the differential system is regularized by a levi civita type transformation and an appropriate scaling of time the initial condition of this orbit is shown to be the infimum of some well chosen set this existence proof is direct and surprisingly simple further a careful study shows that this orbit has a symmetry group isomorphic to the dihedral group d 4 then roberts symmetry reduction method is applied to show the linear stability it turns out that the rhomboidal periodic orbit in the planar equal mass four body problem is linearly stable

robust nonlinear control associating robust feedback linearization and h sub spl infin control in this note a robust nonlinear controller for a nonlinear system subject to model uncertainties is proposed such a controller associates a robust feedback linearization with a robust linear h sub spl infin controller the robust feedback linearization exactly transforms the nonlinear system into a linear system equal to the linear approximation of the original nonlinear system around a nominal operating point the robustness of the resulting overall nonlinear controller is proved by theoretical arguments and illustrated through an application example the control of a magnetic bearing system

constructing nested nodal sets for multivariate polynomial interpolation we present a robust method for choosing multivariate polynomial interpolation nodes our algorithm is an optimization method to greedily minimize a measure of interpolant sensitivity a variant of a weighted lebesgue function nodes are therefore chosen that tend to control oscillations in the resulting interpolant this method can produce an arbitrary number of nodes and is not constrained by the dimension of a complete polynomial space our method is therefore flexible nested nodal sets are produced in spaces of arbitrary dimensions and the number of nodes added at each stage can be arbitrary the algorithm produces a nodal set given a probability measure on the input space thus parameterizing interpolants with respect to finite measures we present examples to show that the method yields nodal sets that behave well with respect to standard interpolation diagnostics the lebesgue constant the vandermonde determinant and the vandermonde condition number we also show that a nongreedy version of the

local stability and performance robustness of nonlinear systems with structured uncertainty this paper focuses on robust stability and performance of uncertain nonlinear systems with structured locally bounded uncertainty it is shown that in a general setting under a considerably mild structural assumption a local small gain condition implies robust local performance the feature of this paper is that the performance is defined in a local sense which is a realistic property inherently nonlinear systems can achieve the local robustness is analyzed purely in terms of input output properties so that the proof involves neither state space descriptions nor hamilton jacobi equations

computing fekete and lebesgue points simplex square disk we have computed point sets with maximal absolute value of the vandermonde determinant fekete points or minimal lebesgue constant lebesgue points on three basic bidimensional compact sets the simplex the square and the disk using routines of the matlab optimization toolbox we have obtained some of the best bivariate interpolation sets known so far

linear stability of double double orbits in the parallelogram four body problem abstract we study the linear stability of a two parameter family of periodic orbits in the parallelogram four body problem this family was numerically found and named as double double orbits by vanderbei a demonstration of such an orbit is shown by introducing new transformations and applying roberts symmetry reduction method the linear stability can be simplified to the calculation of two eigenvalues a global picture of linear stability of this set with respect to the two parameters is given for the first time actually most of the double double orbits are numerically proved to be linearly stable

on the global dynamics of the anisotropic manev problem we study the global flow of the anisotropic manev problem which describes the planar motion of two bodies under the influence of an anisotropic newtonian potential with a relativistic correction term we first find all the heteroclinic orbits between equilibrium solutions then we generalize the poincare melnikov method and use it to prove the existence of infinitely many transversal homoclinic orbits invoking a variational principle and the symmetries of the system we finally detect infinitely many classes of periodic solutions

action minimizing orbits in the parallelogram four body problem with equal masses in this paper we prove the existence of a new periodic solution for the planar newtonian four body problem with equal masses on this orbit two mass points travel on one star shaped closed curve while the other two travel on another and have the opposite orientation the configuration of the masses changes from square to collinear periodically and remains a parallelogram for all time

simple choreographies of the planar newtonian n body problem in the n body problem a simple choreography is a periodic solution where all masses chase each other on a single loop in this paper we prove that for the planar newtonian n body problem with equal masses n 3 there are at least 2n 3 2 n 3 2 different main simple choreographies this confirms a conjecture given by chenciner et al geometry mechanics and dynamics springer new york pp 287 308 2002 all the simple choreoagraphies we prove belong to the linear chain family

morse index and stability of elliptic lagrangian solutions in the planar three body problem abstract we illustrate a new way to study the stability problem in celestial mechanics in this paper using the variational nature of elliptic lagrangian solutions in the planar three body problem we study the relation between morse index and its stability via maslov type index theory of periodic solutions of hamiltonian system for elliptic lagrangian solutions we get an estimate of the algebraic multiplicity of unit eigenvalues of its monodromy matrix in terms of the morse index which is the key to understand the stability problem as a special case we provide a criterion to spectral stability of relative equilibrium

variational proof of the existence of the super eight orbit in the four body problem using the variational method chenciner and montgomery ann math 152 881 901 2000 proved the existence of an eight shaped periodic solution of the planar three body problem with equal masses just after the discovery gerver numerically found a similar periodic solution called super eight in the planar four body problem with equal mass in this paper we prove the existence of the super eight orbit by using the variational method the difficulty of the proof is to eliminate the possibility of collisions in order to solve it we apply the scaling technique established by tanaka ann inst h poincare anal non lineaire 10 215 238 1993 proc am math soc 122 275 284 1994 and investigate the asymptotic behavior of a binary collision

correction to local l sub p stability and local small gain theorem for discrete time systems the author points out that one of the proofs spl sigma d is globally l sub p observable given in section vi of the above paper ibid vol 41 1995 contains an error

quadrature over a pyramid or cube of integrands with a singularity at a vertex a simple transformation is introduced which facilitates the evaluation of integrals over certain square based pyramids or cubes in cases where the integrand has a singularity at a vertex

minimizing periodic orbits with regularizable collisions in the n body problem in the newtonian n body problem there are various subsystems with two degrees of freedom such as the collinear three body problem and the isosceles three body problem after we determine a normal form of the lagrangians of these subsystems we prove the existence of periodic solutions with regularizable collisions for these systems our result includes several examples such as schubart s orbit with or without equal masses among others

bivariate polynomial interpolation on the square at new nodal sets abstract as known the problem of choosing good nodes is a central one in polynomial interpolation while the problem is essentially solved in one dimension all good nodal sequences are asymptotically equidistributed with respect to the arc cosine metric in several variables it still represents a substantially open question in this work we consider new nodal sets for bivariate polynomial interpolation on the square first we consider fast leja points for tensor product interpolation on the other hand for interpolation in p n 2 on the square we experiment four families of points which are asymptotically equidistributed with respect to the dubiner metric which extends to higher dimension the arc cosine metric one of them nicknamed padua points gives numerically a lebesgue constant growing like log square of the degree

gauss green cubature and moment computation over arbitrary geometries we have implemented in matlab a gauss like cubature formula over arbitrary bivariate domains with a piecewise regular boundary which is tracked by splines of maximum degree p spline curvilinear polygons the formula is exact for polynomials of degree at most 2n 1 using n cmn 2 nodes 1 c p m being the total number of points given on the boundary it does not need any decomposition of the domain but relies directly on univariate gauss legendre quadrature via green s integral formula several numerical tests are presented including computation of standard as well as orthogonal moments over a nonstandard planar region

computing approximate fekete points by qr factorizations of vandermonde matrices we propose a numerical method implemented in matlab for computing approximate fekete points on compact multivariate domains it relies on the search of maximum volume submatrices of vandermonde matrices computed on suitable discretization meshes and uses a simple greedy algorithm based on qr factorization with column pivoting the method gives also automatically an algebraic cubature formula provided that the moments of the underlying polynomial basis are known numerical tests are presented for the interval and the square which show that approximate fekete points are well suited for polynomial interpolation and cubature

new periodic solutions for 2n body problems in r3 for newtonian 2n body problems with equal masses in r3 we prove the existence of new noncollision periodic solution such that 2n bodies move on two different closed curves

binary decompositions for planar n body problems and symmetric periodic solutions a binary decomposition for a system of n masses is a way of treating the system as big n atop 2 big binaries with the total action exactly the same as that of the original system by considering binary decompositions we are able to provide effective lower bound estimates for the action of collision paths in several spaces of symmetric loops as applications we use our estimates to prove the existence of some new classes of symmetric periodic solutions for the n body problem

polynomial interpolation and cubature over polygons we have implemented a matlab code to compute discrete extremal sets of fekete and leja type on convex or concave polygons together with the corresponding interpolatory cubature formulas the method works by qr and lu factorizations of rectangular vandermonde matrices on weakly admissible meshes wams of polygons constructed by polygon quadrangulation

geometric weakly admissible meshes discrete least squares approximations and approximate fekete points using the concept of geometric weakly admissible meshes see 2 below together with an algorithm based on the classical qr factorization of matrices we compute efficient points for discrete multivariate least squares approximation and lagrange interpolation

symmetry groups and non planar collisionless action minimizing solutions of the three body problem in three dimensional space periodic and quasi periodic solutions of the n body problem can be found as minimizers of the lagrangian action functional restricted to suitable spaces of symmetric paths the main purpose of this paper is to develop a systematic approach to the equivariant minimization for the three body problem in three dimensional space first we give a finite complete list of symmetry groups fitting to the minimization of the action with the property that any other symmetry group can be reduced to be isomorphic to one of these representatives a second step is to prove that the resulting local and global symmetric action minimizers are always collisionless when they are not already bound to collisions furthermore we prove some results which address the question of whether minimizers are planar or non planar as a consequence of our theory we will give general criteria for a symmetry group to yield planar or homographic minimizers either homographic or not as in the chenciner montgomery eight solution on the other hand we will provide a rigorous proof of the existence of some interesting one parameter families of periodic and quasi periodic non planar orbits these include the choreographic marchal s p12 family with equal masses together with a less symmetric choreographic family which anyway probably coincides with the p12 family

existence and minimizing properties of retrograde orbits to the three body problem with various choices of masses poincare made the first attempt in 1896 on applying variational calculus to the three body problem and observed that collision orbits do not necessarily have higher values of action than classical solutions little progress had been made on resolving this difficulty until a recent breakthrough by chenciner and montgomery afterward variational methods were successfully applied to the jv body problem to construct new classes of solutions in order to avoid collisions the problem is confined to symmetric path spaces and all new planar solutions were constructed under the assumption that some masses are equal a question for the variational approach on planar problems naturally arises are minimizing methods useful only when some masses are identical this article addresses this question for the three body problem for various choices of masses it is proved that there exist infinitely many solutions with a certain topological type called retrograde orbits that minimize the action functional on certain path spaces cases covered in our work include triple stars in retrograde motions double stars with one outer planet and some double stars with one planet orbiting around one primary mass our results largely complement the classical results by the poincare continuation method and conley s geometric approach

a lattice basis reduction approach for the design of finite wordlength fir filters many applications of finite impulse response fir digital filters impose strict format constraints on the filter coefficients such requirements increase the complexity of determining optimal designs for the problem at hand we introduce a fast and efficient method based on the computation of good nodes for polynomial interpolation and euclidean lattice basis reduction experiments show that it returns quasi optimal finite wordlength fir filters compared to previous approaches it also scales remarkably well length 125 filters are treated in 9 s it also proves useful for accelerating the determination of optimal finite wordlength fir filters

bounded input bounded output stabilization of nonlinear systems using state detectors abstract this paper is devoted to the study of relations between lyapunov stabilization and bounded input bounded output bibo stabilization of nonlinear systems when the state is not available for measurement and various state detectors are employed for implementing feedback control it is found that with some constraints imposed on the detectors lyapunov stabilization implies bibo stabilization a feedback control scheme for achieving bibo stability is presented

polynomial fitting and interpolation on circular sections we construct weakly admissible polynomial meshes wams on circular sections such as symmetric and asymmetric circular sectors circular segments zones lenses and lunes the construction resorts to recent results on subperiodic trigonometric interpolation the paper is accompanied by a software package to perform polynomial fitting and interpolation at discrete extremal sets on such regions

a stochastic collocation algorithm with multifidelity models we present a numerical method for utilizing stochastic models with differing fideli ties to approximate parameterized functions a representative case is where a high fidelity and a low fidelity model are available the low fidelity model represents a coarse and rather crude ap proximation to the underlying physical system however it is easy to compute and consumes little simulation time on the other hand the high fidelity model is a much more accurate representation of the physics but can be highly time consuming to simulate our approach is nonintrusive and is therefore applicable to stochastic collocation settings where the parameters are random variables we provide sufficient conditions for convergence of the method and present several examples that are of practical interest including multifidelity approximations and dimensionality reduction

polynomial approximation and cubature at approximate fekete and leja points of the cylinder the paper deals with polynomial interpolation least square approximation and cubature of functions dened on the rectangular cylinder k d 1 1 with d the unit disk the nodes used for these processes are the approximate fekete points afp and the discrete leja points dlp extracted from suitable weakly admissible meshes wams of the cylinder from the analysis of the growth of the lebesgue constants approximation and cubature errors we show that the afp and the dlp extracted from wam are good points for polynomial approximation and numerical integration of functions dened on the cylinder

bivariate lagrange interpolation at the padua points the ideal theory approach the padua points are a family of points on the square 1 1 2 given by explicit formulas that admits unique lagrange interpolation by bivariate polynomials interpolation polynomials and cubature formulas based on the padua points are studied from an ideal theoretic point of view which leads to the discovery of a compact formula for the interpolation polynomials the lp convergence of the interpolation polynomials is also studied

saari s conjecture is true for generic vector fields the simplest non collision solutions of the n body problem are the relative equilibria in which each body follows a circular orbit around the centre of mass and the shape formed by the n bodies is constant it is easy to see that the moment of inertia of such a solution is constant in 1970 d saari conjectured that the converse is also true for the planar newtonian n body problem relative equilibria are the only constant inertia solutions a computer assisted proof for the 3 body case was recently given by r moeckel trans amer math soc 2005 we present a different kind of answer proofs that several generalisations of saari s conjecture are generically true our main tool is jet transversality including a new version suitable for the study of generic potential functions

determining passivity using linearization for systems with feedthrough terms in this technical note we consider the following problem what passivity properties for a nonlinear system can be inferred when its linearization around an equilibrium point is known to be passive we consider both continuous time and discrete time systems with feedthrough terms our main results show that when the linearized model is simultaneously strictly passive and strictly input passive the nonlinear system is passive as well within a neighborhood of the equilibrium point around which the linearization is done

bivariate lagrange interpolation at the padua points computational aspects the so called padua points give a simple geometric and explicit construction of bivariate polynomial interpolation in the square moreover the associated lebesgue constant has minimal order of growth o log 2 n here we show four families of padua points for interpolation at any even or odd degree n and we present a stable and efficient implementation of the corresponding lagrange interpolation formula based on the representation in a suitable orthogonal basis we also discuss extension of non polynomial padua like interpolation to other domains such as triangles and ellipses we give complexity and error estimates and several numerical tests

on the existence of collisionless equivariant minimizers for the classical n body problem we show that the minimization of the lagrangian action functional on suitable classes of symmetric loops yields collisionless periodic orbits of the n body problem provided that some simple conditions on the symmetry group are satisfied more precisely we give a fairly general condition on symmetry groups g of the loop space for the n body problem with potential of homogeneous degree with 0 which ensures that the restriction of the lagrangian action mathcal a to the space g of g equivariant loops is coercive and its minimizers are collisionless without any strong force assumption in proving that local minima of g are free of collisions we develop an averaging technique based on marchal s idea of replacing some of the point masses with suitable shapes see 10 as an application several new orbits can be found with some appropriate choice of g furthermore the result can be used to give a simplified and unitary proof of the existence of many already known minimizing periodic orbits

variational property of periodic kepler orbits in constant curvature spaces abstract we study the variational property of the periodic kepler orbits on the sphere the plane and the hyperbolic plane we first classify the orbits by the two constants of motion the energy and the angular momentum then we characterize the local variational property of the closed orbits by computing the maslov type indices finally we study the global variational property of the closed orbits we prove that the closed orbits on the hyperbolic plane minimizes the action among all loops which encircle the attracting center

on selecting a maximum volume sub matrix of a matrix and related problems given a matrix email protected r m x n n vectors in m dimensions we consider the problem of selecting a subset of its columns such that its elements are as linearly independent as possible this notion turned out to be important in low rank approximations to matrices and rank revealing qr factorizations which have been investigated in the linear algebra community and can be quantified in a few different ways in this paper from a complexity theoretic point of view we propose four related problems in which we try to find a sub matrix email protected r m x k of a given matrix email protected r m x n such that i s m a x c the largest singular value of c is minimum ii s m i n c the smallest singular value of c is maximum iii k c email protected m a x c s m i n c the condition number of c is minimum and iv the volume of the parallelepiped defined by the column vectors of c is maximum we establish the np hardness of these problems and further show that they do not admit ptas we then study a natural greedy heuristic for the maximum volume problem and show that it has approximation ratio 2 o k l o g k our analysis of the greedy heuristic is tight to within a logarithmic factor in the exponent which we show by explicitly constructing an instance for which the greedy heuristic is 2 w k from optimal when a has unit norm columns a related problem is to select the maximum number of vectors with a given volume we show that if the optimal solution selects k columns then greedy will select w k logk columns providing a logk approximation

a new branch of mountain pass solutions for the choreographical 3 body problem we prove the existence of a new branch of solutions of mountain pass type for the periodic 3 body problem with choreographical constraint at first we describe the variational structure of the action functional associated to the choreographical three body problem in mathbb r 3 in the second part using a bisection algorithm we provide a numerical non rigorous solution of mountain pass type for this problem in a rotating frame with angular velocity 1 5 the last step consists in the rigorous computer assisted proof of the existence of a full branch of solutions for the problem starting from the mountain pass solution detected numerically

tensor product gauss lobatto points are fekete points for the cube tensor products of gauss lobatto quadrature points are frequently used as collocation points in spectral element methods unfortunately it is not known if gauss lobatto points exist in non tensor product domains like the simplex in this work we show that the n dimensional tensor product of gauss lobatto quadrature points are also fekete points this suggests a way to generalize spectral methods based on gauss lobatto points to non tensor product domains since fekete points are known to exist and have been computed in the triangle and tetrahedron in one dimension this result was proved by fejer in 1932 but the extension to higher dimensions in non trivial

linear stability analysis of symmetric periodic simultaneous binary collision orbits in the planar pairwise symmetric four body problem abstract we apply the symmetry reduction method of roberts to numerically analyze the linear stability of a one parameter family of symmetric periodic orbits with regularizable simultaneous binary collisions in the planar pairwise symmetric four body problem with a mass m 0 1 as the parameter this reduces the linear stability analysis to the computation of two eigenvalues of a 3 3 matrix for each m 0 1 obtained from numerical integration of the linearized regularized equations along only the first one eighth of each regularized periodic orbit the results are that the family of symmetric periodic orbits with regularizable simultaneous binary collisions changes its linear stability type several times as m varies over 0 1 with linear instability for m close or equal to 0 01 and linear stability for m close or equal to 1

a separating surface for sitnikov like n 1 body problems abstract we consider the generalized sitnikov problem of newtonian mechanics for periodic planar configurations of n bodies which are symmetric under rotation by a fixed angle the z axis is invariant we consider the effect of placing a massless particle on the z axis the study of the motion of this particle can then be modeled as a time dependent hamiltonian system we give a geometric construction of a surface in the three dimensional phase space separating orbits for which the massless particle escapes to infinity from those for which it does not a procedure for removing the periodicity condition of the planar configuration is outlined the construction of the surface is demonstrated numerically in a few examples

least squares polynomial approximation on weakly admissible meshes disk and triangle we construct symmetric polar wams weakly admissible meshes with low cardinality for least squares polynomial approximation on the disk these are then mapped to an arbitrary triangle numerical tests show that the growth of the least squares projection uniform norm is much slower than the theoretical bound and even slower than that of the lebesgue constant of the best known interpolation points for the triangle as opposed to good interpolation points such meshes are straightforward to compute for any degree the construction can be extended to polygons by triangulation

computing multivariate fekete and leja points by numerical linear algebra we discuss and compare two greedy algorithms that compute discrete versions of fekete like points for multivariate compact sets by basic tools of numerical linear algebra the first gives the so called approximate fekete points by qr factorization with column pivoting of vandermonde like matrices the second computes discrete leja points by lu factorization with row pivoting moreover we study the asymptotic distribution of such points when they are extracted from weakly admissible meshes

continuation of periodic orbits in conservative and hamiltonian systems we introduce and justify a computational scheme for the continuation of periodic orbits in systems with one or more first integrals and in particular in hamiltonian systems having several independent symmetries our method is based on a generalization of the concept of a normal periodic orbit as introduced by sepulchre and mackay nonlinearity 10 1997 679 we illustrate the continuation method on some integrable hamiltonian systems with two degrees of freedom and briefly discuss some further applications

choreographic three bodies on the lemniscate we show that choreographic three bodies x t x t t 3 x t t 3 of period t on the lemniscate x t cn t sn t 1 cn2 t parametrized by the jacobian elliptic functions sn and cn with modulus k2 2 3 4 conserve the centre of mass and the angular momentum where and are the orthogonal unit vectors defining the plane of the motion they also conserve the moment of inertia the kinetic energy the sum of squares of the curvature the product of distances and the sum of squares of distances between bodies we find that they satisfy the equation of motion under the potential energy i j 1 2 ln rij 3 24 rij2 or i j 1 2 ln rij i 3 8 ri2 where rij is the distance between bodies i and j and ri the distance from the origin the first term of the potential energies is the universal gravitation in two dimensions but the second term is a mutual repulsive force or a repulsive force from the origin respectively then geometric construction methods for the positions of the choreographic three bodies are given