Numerical studies of hyperbolic manifolds supporting diffusion in symplectic mappings

Diffusion in generic quasi integrable systems at small values of the perturbing parameters has been a very studied subject since the pioneering work of Arnold [3]. For moderate values of the perturbing parameter a different kind of diffusion occurs, the so called Chirikov diffusion, since the Chirikov’s papers [11, 13]. The two underlying mechanisms are different, the first has an analytic demonstration only on specific models, the second is based on an heuristic argument. Even if the relation between chaos and diffusion is far to be completely understood, a key role is played by the topology of hyperbolic manifolds related to the resonances. Different methods can be found in the literature for the detection of hyperbolic manifolds, at least for two dimensional systems. For higher dimensional ones some sophisticated methods have been recently developed (for a review see [55]). In this paper we review some of these methods and an easy tool of detection of invariant manifolds that we have developed based on the Fast Lyapunov Indicator. The relation between the topology of hyperbolic manifolds and diffusion is discussed in the framework of Arnold diffusion.     

Solvability of a Commutative Algebra which Satisfies (x 2)2 = 0

We studied the solvability of the algebra which satisfies the polynomial identity (x ²)² = 0. We believe that, if A is a finite dimensional commutative algebra over a field F of characteristic not 2 which satisfies (x ²)² = 0 for all x ∈ A, then A is solvable. In this article we proved this when dim  _F A ≤ 7.     

Convergence to the Time Average by Stochastic Regularization

In Ergodic Theory it is natural to consider the pointwise convergence of finite time averages of functions with respect to the flow of dynamical systems. Since the pointwise convergence is too weak for applications to Hamiltonian Perturbation Theory, requiring differentiability, we first introduce regularized averages obtained through a stochastic perturbation of an integrable Hamiltonian flow, and then we provide detailed estimates. In particular, for a special vanishing limit of the stochastic perturbation, we obtain convergence even in a Sobolev norm taking into account the derivatives.     

A Numerical Study of Arnold Diffusion in a Priori Unstable Systems

This paper concerns the problem of the numerical detection of Arnold diffusion in a priori unstable systems. Specifically, we introduce a new definition of Arnold diffusion which is adapted to the numerical investigation of the problem, and is based on the numerical computation of the stable and unstable manifolds of the system. Examples of this Arnold diffusion are provided in a model system. In this model, we also find that Arnold diffusion behaves as an approximate Markovian process, thus it becomes possible to compute diffusion coefficients. The values of the diffusion coefficients satisfy the scaling . We also find that this law is correlated to the validity of the Melnikov approximation: in fact, the law is valid up to the same critical value of ε for which the error terms of Melnikov approximations have a sharp increment.     

A numerical study of the topology of normally hyperbolic invariant manifolds supporting Arnold diffusion in quasi-integrable systems

We investigate numerically the stable and unstable manifolds of the hyperbolic manifolds of the phase space related to the resonances of quasi-integrable systems in the regime of validity of the Nekhoroshev and KAM theorems. Using a model of weakly interacting resonances we explain the qualitative features of these manifolds characterized by peculiar ‘flower-like’ structures. We detect different transitions in the topology of these manifolds related to the local rational approximations of the frequencies. We find numerically a correlation among these transitions and the speed of Arnold diffusion.     

An accelerated covering relaxation algorithm for solving 0–1 positive polynomial programs

The purpose of this note is to present an accelerated algorithm for solving 0–1 positive polynomial (PP) problems. Like our covering relaxation algorithm (Management Science 1979), the accelerated algorithm is a cutting plane method, which uses the linear set covering problem as a relaxation for PP. However, a unique and novel feature of the accelerated algorithm is that it attempts to generate cutting planes from heuristic solutions to the set covering problem whenever possible. Computational results reveal that this strategy of generating cutting planes has led to a significant reduction in the computational time required to solve a PP problem.This research was partially supported by National Sciences and Engineering Research Council Canada Grants 67-4181 and 67-3998, Office of Naval Research Contract N00014-76-C-0418, and National Science Foundation Grant ECS80-22027.     

Analysis of 10 urinary BTEX metabolites using LC–MS/MS

Benzene, toluene, ethylbenzene, and xylene (BTEX) are a group of volatile organic compounds that are ubiquitous in the environment due to numerous anthropogenic sources. Exposure to BTEX poses a health hazard by increasing the risk for damage to multiple organs, neurocognitive impairment and birth defects. Urinary BTEX metabolites are useful biomarkers for the evaluation of BTEX exposure, because of the ease of sampling and their longer physiological half-lives compared with parent compounds. A method that utilizes LC–MS/MS was developed and validated for simultaneously monitoring of 10 urinary BTEX metabolites. During the sample preparation an aliquot of urine was diluted with an equal volume of 1% formic acid; internal standard solution was added, and then the sample was centrifuged and analyzed. The analytes were separated on the Kinetex-F5 column by applying a linear gradient, consisting of 0.1% formic acid and methanol. The method was validated according to the FDA Bioanalytical Method Validation Guidance for Industry. The mean method's accuracies of the spiked matrix were 81–122%; the inter-day precision ranged from 4 to 20%; the limits of quantitation were 0.5–2 μg/L. The method was used for the evaluation of baseline levels of urinary BTEX metabolites in 87 firefighters.