Symplectic phase flow approximation for the numerical integration of canonical systems

Summary New methods are presented for the numerical integration of ordinary differential equations of the important family of Hamiltonian dynamical systems. These methods preserve the Poincaré invariants and, therefore, mimic relevant qualitative properties of the exact solutions. The methods are based on a Runge-Kutta-type ansatz for the generating function to realize the integration steps by canonical transformations. A fourth-order method is given and its implementation is discussed. Numerical results are presented for the Hénon-Heiles system, which describes the motion of a star in an axisymmetric galaxy.     

KAM Stability for a three-body problem of the Solar system

A new (iso-energetic) KAM method is tested on a specific three-body problem “extracted” from the Solar system (Sun-Jupiter + asteroid 12 Victoria). Analytical results in agreement with the observed data are established. This paper is a concise presentation of [2]. Supported by the MIUR projects: “Dynamical Systems: Classical, Quantum, Stochastic” and “Variational Methods and Nonlinear Differential Equations” Received: February 3, 2004     

Estimation for stochastic damping hamiltonian systems under partial observation—I. Invariant density

In this paper, we study the non-parametric estimation of the invariant density of some ergodic hamiltonian systems, using kernel estimators. The main result is a central limit theorem for such estimators under partial observation (only the positions are observed). The main tools are mixing estimates and refined covariance inequalities, the main difficulty being the strong degeneracy of such processes. This is the first paper of a series of at least two, devoted to the estimation of the characteristics of such processes: invariant density, drift term, volatility.     

Error expansion for the discretization of backward stochastic differential equations

We study the error induced by the time discretization of decoupled forward–backward stochastic differential equations (X,Y,Z)$(X,Y,Z)$. The forward component X$X$ is the solution of a Brownian stochastic differential equation and is approximated by a Euler scheme X^N$X^N$ with N$N$ time steps. The backward component is approximated by a backward scheme. Firstly, we prove that the errors (Y^N−Y,Z^N−Z)$(Y^N-Y,Z^N-Z)$ measured in the strong _Lp$L_p$-sense (p≥1$p\ge 1$) are of order N^−1/2$N^{-1/2}$ (this generalizes the results by Zhang [J. Zhang, A numerical scheme for BSDEs, The Annals of Applied Probability 14 (1) (2004) 459–488]). Secondly, an error expansion is derived: surprisingly, the first term is proportional to X^N−X$X^N-X$ while residual terms are of order N⁻¹$N^{-1}$.     

A Central Limit Theorem for isotropic flows

We establish that the image of a measure, which satisfies a certain energy condition, moving under a standard isotropic Brownian flow will, when properly scaled, have an asymptotically normal distribution under almost every realization of the flow. We derive the same result for an initial point mass moved by an isotropic Kraichnan flow.     

A Numerical Method for the Dynamical Correction of Celestial Reference Frames from Non-Regular Samples

The aim of this paper is to analyze the orientation errors of a celestial reference system. These errors can be obtained by analyzing the differences between the observed and calculated positions for a set of selected minor planets. The classical methods do not work well if the sample is non-homogeneous on the region covered by the observations. In this paper a new algorithm based on a new class of spatial estimators and an appropriate reconstruction operator is proposed.     

Power integral bases for Selmer-like number fields

The Selmer trinomials are the trinomials f(X)∈{Xn−X−1,Xn+X+1|n>1 is an integer} over Z. For these trinomials we show that the ideal C=(f(X),f^′(X))Z[X] has height two and contains the linear polynomial (n−1)X+n. We then give several necessary and sufficient conditions for D[X]/(f(X)D[X]) to be a regular ring, where f(X) is an arbitrary polynomial over a Dedekind domain D such that its ideal C has height two and contains a product of primitive linear polynomials. We next specialize to the Selmer-like trinomials bXn+cX+d and bXn+cXn−1+d over D and give several more such necessary and sufficient conditions (among them is that C is a radical ideal). We then specialize to the Selmer trinomials over Z and give quite a few more such conditions (among them is that the discriminant Disc(Xn−X−1)=±(nn−(1−n)ⁿ⁻¹) of Xn−X−1 is square-free (respectively Disc(Xn+X+1)=±(nn+(1−n)ⁿ⁻¹) of Xn+X+1 is square-free)). Finally, we show that nn+(1−n)ⁿ⁻¹ is never square-free when n≡2 (mod 3) and n>2, but, otherwise, both are very often (but not always) square-free.     

Convergence and stability of implicit runge-kutta methods for systems with multiplicative noise

A class ofimplicit Runge-Kutta schemes for stochastic differential equations affected bymultiplicative Gaussian white noise is shown to be optimal with respect to global order of convergence in quadratic mean. A test equation is proposed in order to investigate the stability of discretization methods for systems of this kind. Herestability is intended in a truly probabilistic sense, as opposed to the recently introduced extension of A-stability to the stochastic context, given for systems with additive noise. Stability regions for the optimal class are also given.Partially supported by the Italian Consiglio Nazionale delle Ricerche.     

A continuum of periodic solutions to the planar four-body problem with two pairs of equal masses

In this paper, we apply the variational method with Structural Prescribed Boundary Conditions (SPBC) to prove the existence of periodic and quasi-periodic solutions for the planar four-body problem with two pairs of equal masses $m_1=m_3$ and $m_2=m_4$. A path $q(t)$ on $[0,T]$ satisfies the SPBC if the boundaries $q(0)\in \mathbf{A}$ and $q(T)\in \mathbf{B}$, where A and B are two structural configuration spaces in ${({\mathbf{R}}^2)}^4$ and they depend on a rotation angle $\theta \in (0,2\pi )$ and the mass ratio $\mu =\frac{m_2}{m_1}\in {\mathbf{R}}^{+}$.We show that there is a region $\mathrm{\Omega }?(0,2\pi )\times R^{+}$ such that there exists at least one local minimizer of the Lagrangian action functional on the path space satisfying the SPBC $\{q(t)\in H^1([0,T],{({\mathbf{R}}^2)}^4)|q(0)\in \mathbf{A},q(T)\in \mathbf{B}\}$ for any $(\theta ,\mu )\in \mathrm{\Omega }$. The corresponding minimizing path of the minimizer can be extended to a non-homographic periodic solution if θ is commensurable with π or a quasi-periodic solution if θ is not commensurable with π. In the variational method with the SPBC, we only impose constraints on the boundary and we do not impose any symmetry constraint on solutions. Instead, we prove that our solutions that are extended from the initial minimizing paths possess certain symmetries.The periodic solutions can be further classified as simple choreographic solutions, double choreographic solutions and non-choreographic solutions. Among the many stable simple choreographic orbits, the most extraordinary one is the stable star pentagon choreographic solution when $(\theta ,\mu )=(\frac{4\pi }{5},1)$. Remarkably the unequal-mass variants of the stable star pentagon are just as stable as the equal mass choreographies.     

Coherence for modalities

Positive modalities in S4, S5 and systems in their vicinity are investigated in terms of categorial proof theory. Coherence and maximality results are demonstrated, and connections with mixed distributive laws and Frobenius algebras are exhibited.     

Asymptotic properties of maximum likelihood estimator for the growth rate for a jump-type CIR process based on continuous time observations

We consider a jump-type Cox–Ingersoll–Ross (CIR) process driven by a standard Wiener process and a subordinator, and we study asymptotic properties of the maximum likelihood estimator (MLE) for its growth rate. We distinguish three cases: subcritical, critical and supercritical. In the subcritical case we prove weak consistency and asymptotic normality, and, under an additional moment assumption, strong consistency as well. In the supercritical case, we prove strong consistency and mixed normal (but non-normal) asymptotic behavior, while in the critical case, weak consistency and non-standard asymptotic behavior are described. We specialize our results to so-called basic affine jump–diffusions as well. Concerning the asymptotic behavior of the MLE in the supercritical case, we derive a stochastic representation of the limiting mixed normal distribution, where the almost sure limit of an appropriately scaled jump-type supercritical CIR process comes into play. This is a new phenomenon, compared to the critical case, where a diffusion-type critical CIR process plays a role.     

Sparse approximate inverse preconditioning for dense linear systems arising in computational electromagnetics

We investigate the use of sparse approximate inverse preconditioners for the iterative solution of linear systems with dense complex coefficient matrices arising in industrial electromagnetic problems. An approximate inverse is computed via a Frobenius norm approach with a prescribed nonzero pattern. Some strategies for determining the nonzero pattern of an approximate inverse are described. The results of numerical experiments suggest that sparse approximate inverse preconditioning is a viable approach for the solution of large-scale dense linear systems on parallel computers. This revised version was published online in August 2006 with corrections to the Cover Date.     

On Fictitious Domain Formulations for Maxwell''s Equations

We consider fictitious domain-Lagrange multiplier formulations for variational problems in the space H(curl: Ω{\bf)} derived from Maxwell's equations. Boundary conditions and the divergence constraint are imposed weakly by using Lagrange multipliers. Both the time dependent and time harmonic formulations of the Maxwell's equations are considered, and we derive well-posed formulations for both cases. The variational problem that arises can be discretized by functions that do not satisfy an a-priori divergence constraint.     

Some limit theorems for the eigenvalues of a sample covariance matrix

Limit theorems are given for the eigenvalues of a sample covariance matrix when the dimension of the matrix as well as the sample size tend to infinity. The limit of the cumulative distribution function of the eigenvalues is determined by use of a method of moments. The proof is mainly combinatorial. By a variant of the method of moments it is shown that the sum of the eigenvalues, raised to k-th power, k = 1, 2,…, m is asymptotically normal. A limit theorem for the log sum of the eigenvalues is completed with estimates of expected value and variance and with bounds of Berry-Esseen type.     

Limit theorems for multivariate Bessel processes in the freezing regime

Multivariate Bessel processes describe the stochastic dynamics of interacting particle systems of Calogero–Moser–Sutherland type and are related with $\beta$-Hermite and Laguerre ensembles. It was shown by Andraus, Katori, and Miyashita that for fixed starting points, these processes admit interesting limit laws when the multiplicities $k$ tend to $\infty$, where in some cases the limits are described by the zeros of classical Hermite and Laguerre polynomials. In this paper we use SDEs to derive corresponding limit laws for starting points of the form $\sqrt{k}?x$ for $k\to \infty$ with $x$ in the interior of the corresponding Weyl chambers. Our limit results are a.s. locally uniform in time. Moreover, in some cases we present associated central limit theorems.