A class of symplectic integrators with adaptive timestep for separable Hamiltonian systems

Symplectic integration algorithms are well-suited for long-term integrations of Hamiltonian systems because they preserve the geometric structure of the Hamiltonian flow. However, this desirable property is generally lost when adaptive timestep control is added to a symplectic integrator. We describe an adaptive-timestep symplectic integrator that can be used if the Hamiltonian is the sum of kinetic and potential energy components and the required timestep depends only on the potential energy (e.g. test-particle integrations in fixed potentials). In particular, we describe an explicit, reversible, symplectic, leapfrog integrator for a test particle in a near-Keplerian potential; this integrator has timestep proportional to distance from the attracting mass and has the remarkable property of integrating orbits in an inverse-square force field with only "along-track" errors; i.e. the phase-space shape of a Keplerian orbit is reproduced exactly, but the orbital period is in error by O(1/N^2), where N is the number of steps per period.Comment: 24 pages, 3 figures, submitted to Astronomical Journal; minor errors in equations and one figure correcte

Corporate Ownership Structure and the Evolution of Bankruptcy Law: Lessons from the United Kingdom

The corporate world today subdivides into rival systems of dispersed and concentrated ownership, each characterized by different corporate governance structures. The United States falls into the former category, whereas major industrial rivals such as Japan and Germany are members of the latter. The past decade has seen intense academic debate over possible explanations for the different systems of ownership and control in key developed economies. Anecdotal evidence suggesting that market forces may be serving to destabilize traditional business structures and foster some form of convergence in a U.S. direction has given the controversy powerful current relevance. For those seeking to account for the existence of rival systems of dispersed and concentrated ownership, the United Kingdom has proved to be something of a problem child. Britain is a companion to the United States in the dispersed ownership category since, as is the case in the U.S., publicly quoted companies are a pivotal feature of the corporate economy, and large business enterprises typically have diffuse ownership structures. Given the similarities between the two countries, a logical way to test the various theories offered to account for the configuration of America\u27s system of ownership and control is to see whether they have explanatory power in a British context. When this sort of analysis has been done, however, events occurring in the U.K. have tended to cast doubt upon each hypothesis. This has been the outcome, for instance, with theories concerning financial services regulation, political ideology, and minority shareholder protection. This Article\u27s purpose is to refer again to the British experience to test and refine an additional hypothesis that has been offered to explain why the corporate economy is organized differently in the U.S. than in countries such as Germany and Japan. Corporate bankruptcy, it has been said, is the crucial missing piece in understanding corporate governance. According to this thesis-an evolutionary account of corporate governance-a country\u27s system of bankruptcy law is either manager-driven or manager-displacing, with the former offering the executives of a financially troubled firm substantial scope to launch a rescue effort and the latter having a strong bias in favor of liquidation. The thinking, in very basic terms, is that a manager-driven bankruptcy regime complements dispersed share ownership, while its manager-displacing counterpart aligns with a governance regime where concentrated ownership prevails. Given the configuration of the U.K.\u27s system of ownership and control, this would imply that Britain should have a manager-driven bankruptcy system. As we will see, though, the country\u27s bankruptcy laws strongly protect lenders, and few British companies that end up in formal bankruptcy proceedings escape liquidation

Exact evolution of time-reversible symplectic integrators and their phase error for the harmonic oscillator

The evolution of any factorized time-reversible symplectic integrators, when applied to the harmonic oscillator, can be exactly solved in a closed form. The resulting modified Hamiltonians demonstrate the convergence of the Lie series expansions. They are also less distorted than modified Hamiltonian of non-reversible algorithms. The analytical form for the modified angular frequency can be used to assess the phase error of any time-reversible algorithm.Comment: Submitted to Phys. Lett. A, Six Pages two Column

Corporate Bankruptcy Panel--The Fraudulent Conveyance Origins of Chapter 11: An Essay on the Unwritten Law of Corporate Reorganizations

The second panel was the Corporate Panel, led by Professor Douglas Baird from the University of Chicago Law School. We were fortunate enough to have the impressive Sarah R. Borders act as the moderator of this panel. The discussion, however, did not require much moderating; the panelists were familiar with one another and the discussion was entertaining. The panel was divided into two schools of thought: those who felt like the Code solely governed the practice of corporate bankruptcy, and those who felt like there was a set of unwritten rules created over the last 150 years that pulled the strings

A Hybrid N-body--Coagulation Code for Planet Formation

We describe a hybrid algorithm to calculate the formation of planets from an initial ensemble of planetesimals. The algorithm uses a coagulation code to treat the growth of planetesimals into oligarchs and explicit N-body calculations to follow the evolution of oligarchs into planets. To validate the N-body portion of the algorithm, we use a battery of tests in planetary dynamics. Several complete calculations of terrestrial planet formation with the hybrid code yield good agreement with previously published calculations. These results demonstrate that the hybrid code provides an accurate treatment of the evolution of planetesimals into planets.Comment: Astronomical Journal, accepted; 33 pages + 11 figure

On an asymptotic method for computing the modified energy for symplectic methods

We revisit an algorithm by Skeel et al. [5,16] for computing the modified, or shadow, energy associated with symplectic discretizations of Hamiltonian systems. We amend the algorithm to use Richardson extrapolation in order to obtain arbitrarily high order of accuracy. Error estimates show that the new method captures the exponentially small drift associated with such discretizations. Several numerical examples illustrate the theory

Improving corporate governance in state-owned corporations in China: which way forward?

This article discusses corporate governance in China. It outlines the basic agency problem in Chinese listed companies and questions the effectiveness of the current mechanisms employed to improve their standards of governance. Importantly, it considers alternative means through which corporate practice in China can be brought into line with international expectations and stresses the urgency with which this task must be tackled. It concludes that regulators in China must construct a corporate governance model which is compatible with its domestic setting and not rush to adopt governance initiatives modelled on those in cultures which are fundamentally different in the hope of also reproducing their success

Evaluation of Rehabilitation of Memory in Neurological Disabilities (ReMiND): a randomized controlled trial

OBJECTIVE:The evidence for the effectiveness of memory rehabilitation is inconclusive. The aim was to compare the effectiveness of two group memory rehabilitation programmes with a self-help group control. DESIGN:Single-blind randomized controlled trial. PARTICIPANTS:Participants with memory problems following traumatic brain injury, stroke or multiple sclerosis were recruited from community settings. INTERVENTIONS:Participants were randomly allocated, in cohorts of four, to compensation or restitution group treatment programmes or a self-help group control. All programmes were manual-based and comprised two individual and ten weekly group sessions. MAIN MEASURES:Memory functions, mood, and activities of daily living were assessed at baseline and five and seven months after randomization. RESULTS:There were 72 participants (mean age 47.7, SD 10.2 years; 32 men). There was no significant effect of treatment on the Everyday Memory Questionnaire (P = 0.97). At seven months the mean scores were comparable (restitution 36.6, compensation 41.0, self-help 44.1). However, there was a significant difference between groups on the Internal Memory Aids Questionnaire (P = 0.002). The compensation and restitution groups each used significantly more internal memory aids than the self-help group (P 0.05). CONCLUSIONS:There results show few statistically significant effects of either compensation or restitution memory group treatment as compared with a self-help group control. Further randomized trials of memory rehabilitation are needed