Pseudo-High-Order Symplectic Integrators

Symplectic N-body integrators are widely used to study problems in celestial mechanics. The most popular algorithms are of 2nd and 4th order, requiring 2 and 6 substeps per timestep, respectively. The number of substeps increases rapidly with order in timestep, rendering higher-order methods impractical. However, symplectic integrators are often applied to systems in which perturbations between bodies are a small factor of the force due to a dominant central mass. In this case, it is possible to create optimized symplectic algorithms that require fewer substeps per timestep. This is achieved by only considering error terms of order epsilon, and neglecting those of order epsilon^2, epsilon^3 etc. Here we devise symplectic algorithms with 4 and 6 substeps per step which effectively behave as 4th and 6th-order integrators when epsilon is small. These algorithms are more efficient than the usual 2nd and 4th-order methods when applied to planetary systems.Comment: 14 pages, 5 figures. Accepted for publication in the Astronomical Journa

The role of chaotic resonances in the solar system

Our understanding of the Solar System has been revolutionized over the past decade by the finding that the orbits of the planets are inherently chaotic. In extreme cases, chaotic motions can change the relative positions of the planets around stars, and even eject a planet from a system. Moreover, the spin axis of a planet-Earth's spin axis regulates our seasons-may evolve chaotically, with adverse effects on the climates of otherwise biologically interesting planets. Some of the recently discovered extrasolar planetary systems contain multiple planets, and it is likely that some of these are chaotic as well.Comment: 28 pages, 9 figure

Chirikov Diffusion in the Asteroidal Three-Body Resonance (5,-2,-2)

The theory of diffusion in many-dimensional Hamiltonian system is applied to asteroidal dynamics. The general formulations developed by Chirikov is applied to the Nesvorn\'{y}-Morbidelli analytic model of three-body (three-orbit) mean-motion resonances (Jupiter-Saturn-asteroid system). In particular, we investigate the diffusion \emph{along} and \emph{across} the separatrices of the (5,-2,-2) resonance of the (490) Veritas asteroidal family and their relationship to diffusion in semi-major axis and eccentricity. The estimations of diffusion were obtained using the Melnikov integral, a Hadjidemetriou-type sympletic map and numerical integrations for times up to $10^{8}$ years.Comment: 27 pages, 6 figure

Evidence for an extended scattered disk

By telescopic tracking, we have established that the orbit of the trans-neptunian object (2000 CR$_{105}$) has a perihelion of $\simeq$44 AU, and is thus outside the domain controlled by strong gravitational close encounters with Neptune. Because this object is on a very large, eccentric orbit (with semimajor axis $a\simeq$216 AU and eccentricity $e\simeq$0.8) this object must have been placed on this orbit by a gravitational perturbation which is {\it not} direct gravitational scattering off of any of the giant planets (on their current orbits). The existence of this object may thus have profound cosmogonic implications for our understanding of the formation of the outer Solar System. We discuss some viable scenarios which could have produced it, including long-term diffusive chaos and scattering off of other massive bodies in the outer Solar System. This discovery implies that there must be a large population of trans-neptunian objects in an `extended scattered disk' with perihelia above the previously-discussed 38 AU boundary.Comment: 22 test pages, 4 figures, 2 tables. Submitted to Icarus, 26 March 200

An Overview of the 13:8 Mean Motion Resonance between Venus and Earth

It is known since the seminal study of Laskar (1989) that the inner planetary system is chaotic with respect to its orbits and even escapes are not impossible, although in time scales of billions of years. The aim of this investigation is to locate the orbits of Venus and Earth in phase space, respectively to see how close their orbits are to chaotic motion which would lead to unstable orbits for the inner planets on much shorter time scales. Therefore we did numerical experiments in different dynamical models with different initial conditions -- on one hand the couple Venus-Earth was set close to different mean motion resonances (MMR), and on the other hand Venus' orbital eccentricity (or inclination) was set to values as large as e = 0.36 (i = 40deg). The couple Venus-Earth is almost exactly in the 13:8 mean motion resonance. The stronger acting 8:5 MMR inside, and the 5:3 MMR outside the 13:8 resonance are within a small shift in the Earth's semimajor axis (only 1.5 percent). Especially Mercury is strongly affected by relatively small changes in eccentricity and/or inclination of Venus in these resonances. Even escapes for the innermost planet are possible which may happen quite rapidly.Comment: 14 pages, 11 figures, submitted to CMD

The presence of the casein kinase II phosphorylation sites of Vpu enhances the CD4+ T cell loss caused by the simian–human immunodeficiency virus SHIVKU-lbMC33 in pig-tailed macaques

AbstractThe simian–human immunodeficiency virus (SHIV)/ macaque model for human immunodeficiency virus type 1 has become a useful tool to assess the role of Vpu in lentivirus pathogenesis. In this report, we have mutated the two phosphorylated serine residues of the HIV-1 Vpu to glycine residues and have reconstructed a SHIV expressing this nonphosphorylated Vpu (SHIVS52,56G). Expression studies revealed that this protein was localized to the same intracellular compartment as wild-type Vpu. To determine if this virus was pathogenic, four pig-tailed macaques were inoculated with SHIVS52,56G and virus burdens and circulating CD4+ T cells monitored up to 1 year. Our results indicate that SHIVS52,56G caused rapid loss in the circulating CD4+ T cells within 3 weeks of inoculation in one macaque (CC8X), while the other three macaques developed no or gradual numbers of CD4+ T cells and a wasting syndrome. Histological examination of tissues revealed that macaque CC8X had lesions in lymphoid tissues (spleen, lymph nodes, and thymus) that were typical for macaques inoculated with pathogenic parental SHIVKU-1bMC33 and had no lesions within the CNS. To rule out that macaque CC8X had selected for a virus in which there was reversion of the glycine residues at positions 52 and 56 to serine residues and/or compensating mutations occurred in other genes associated with CD4 down-regulation, sequence analysis was performed on amplified vpu sequences isolated from PBMC and from several lymphoid tissues at necropsy. Sequence analysis revealed a reversion of the glycine residues back to serine residues in this macaque. The other macaques maintained low virus burdens, with one macaque (P003) developing a wasting syndrome between months 9 and 11. Histological examination of tissues from this macaque revealed a thymus with severe atrophy that was similar to that of a previously reported macaque inoculated with a SHIV lacking vpu (Virology 293, 2002, 252). Sequence analysis revealed no reversion of the glycine residues in the vpu sequences isolated from this macaque. These results contrast with those from four macaques inoculated with the parental pathogenic SHIVKU-1bMC33, all of which developed severe CD4+ T cell loss within 1 month after inoculation. Taken together, these results indicate that casein kinase II phosphorylation sites of Vpu contributes to the pathogenicity of the SHIVKU-1bMC33 and suggest that the SHIVKU-1bMC33/pig-tailed macaque model will be useful in analyzing amino acids/domains of Vpu that contribute to the pathogenesis of HIV-1

Mechanical Systems with Symmetry, Variational Principles, and Integration Algorithms

This paper studies variational principles for mechanical systems with symmetry and their applications to integration algorithms. We recall some general features of how to reduce variational principles in the presence of a symmetry group along with general features of integration algorithms for mechanical systems. Then we describe some integration algorithms based directly on variational principles using a discretization technique of Veselov. The general idea for these variational integrators is to directly discretize Hamilton’s principle rather than the equations of motion in a way that preserves the original systems invariants, notably the symplectic form and, via a discrete version of Noether’s theorem, the momentum map. The resulting mechanical integrators are second-order accurate, implicit, symplectic-momentum algorithms. We apply these integrators to the rigid body and the double spherical pendulum to show that the techniques are competitive with existing integrators

A model for collisions in granular gases

We propose a model for collisions between particles of a granular material and calculate the restitution coefficients for the normal and tangential motion as functions of the impact velocity from considerations of dissipative viscoelastic collisions. Existing models of impact with dissipation as well as the classical Hertz impact theory are included in the present model as special cases. We find that the type of collision (smooth, reflecting or sticky) is determined by the impact velocity and by the surface properties of the colliding grains. We observe a rather nontrivial dependence of the tangential restitution coefficient on the impact velocity.Comment: 11 pages, 2 figure

APOE genotype and cognitive change in young, middle-aged, and older adults living in the community.

We examined whether the apolipoprotein E (APOE) ε4 allele was associated with cognitive benefits in young adulthood and whether it reversed to confer cognitive deficits in later life ("antagonistic pleiotropy") in the absence of dementia-related neuropathology. We also tested whether the ε2 allele was associated with disadvantages in early adulthood but offered protection against cognitive decline in early old age. Eight-year cognitive change was assessed in 2,013 cognitively normal community-dwelling adults aged 20-24, 40-44, or 60-64 years at baseline. Although cognitive decline was associated with age, multilevel models contrasting the ε2 and ε4 alleles provided no evidence that the APOE genotype was related to cognitive change in any of the age groups. The findings suggest that in the absence of clinically salient dementia pathology, APOE ε2 and ε4 alleles do not exhibit antagonistic pleiotropy in relation to cognition between the ages of 20 and 72 years

Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond

Many historians of the calculus deny significant continuity between infinitesimal calculus of the 17th century and 20th century developments such as Robinson's theory. Robinson's hyperreals, while providing a consistent theory of infinitesimals, require the resources of modern logic; thus many commentators are comfortable denying a historical continuity. A notable exception is Robinson himself, whose identification with the Leibnizian tradition inspired Lakatos, Laugwitz, and others to consider the history of the infinitesimal in a more favorable light. Inspite of his Leibnizian sympathies, Robinson regards Berkeley's criticisms of the infinitesimal calculus as aptly demonstrating the inconsistency of reasoning with historical infinitesimal magnitudes. We argue that Robinson, among others, overestimates the force of Berkeley's criticisms, by underestimating the mathematical and philosophical resources available to Leibniz. Leibniz's infinitesimals are fictions, not logical fictions, as Ishiguro proposed, but rather pure fictions, like imaginaries, which are not eliminable by some syncategorematic paraphrase. We argue that Leibniz's defense of infinitesimals is more firmly grounded than Berkeley's criticism thereof. We show, moreover, that Leibniz's system for differential calculus was free of logical fallacies. Our argument strengthens the conception of modern infinitesimals as a development of Leibniz's strategy of relating inassignable to assignable quantities by means of his transcendental law of homogeneity.Comment: 69 pages, 3 figure