Forward Symplectic Integrators and the Long Time Phase Error in Periodic Motions

We show that when time-reversible symplectic algorithms are used to solve periodic motions, the energy error after one period is generally two orders higher than that of the algorithm. By use of correctable algorithms, we show that the phase error can also be eliminated two orders higher than that of the integrator. The use of fourth order forward time step integrators can result in sixth order accuracy for the phase error and eighth accuracy in the periodic energy. We study the 1-D harmonic oscillator and the 2-D Kepler problem in great details, and compare the effectiveness of some recent fourth order algorithms.Comment: Submitted to Phys. Rev. E, 29 Page

Symplectic integrators with adaptive time steps

In recent decades, there have been many attempts to construct symplectic integrators with variable time steps, with rather disappointing results. In this paper we identify the causes for this lack of performance, and find that they fall into two categories. In the first, the time step is considered a function of time alone, \Delta=\Delta(t). In this case, backwards error analysis shows that while the algorithms remain symplectic, parametric instabilities arise because of resonance between oscillations of \Delta(t) and the orbital motion. In the second category the time step is a function of phase space variables \Delta=\Delta(q,p). In this case, the system of equations to be solved is analyzed by introducing a new time variable \tau with dt=\Delta(q,p) d\tau. The transformed equations are no longer in Hamiltonian form, and thus are not guaranteed to be stable even when integrated using a method which is symplectic for constant \Delta. We analyze two methods for integrating the transformed equations which do, however, preserve the structure of the original equations. The first is an extended phase space method, which has been successfully used in previous studies of adaptive time step symplectic integrators. The second, novel, method is based on a non-canonical mixed-variable generating function. Numerical trials for both of these methods show good results, without parametric instabilities or spurious growth or damping. It is then shown how to adapt the time step to an error estimate found by backward error analysis, in order to optimize the time-stepping scheme. Numerical results are obtained using this formulation and compared with other time-stepping schemes for the extended phase space symplectic method.Comment: 23 pages, 9 figures, submitted to Plasma Phys. Control. Fusio

Patterns of autism spectrum symptomatology in individuals with Down syndrome without comorbid autism spectrum disorder

Background Prevalence estimates of autism spectrum disorder (ASD) in Down syndrome (DS) are highly varied. This variation is partly due to the difficulty of screening for and diagnosing comorbid ASD in individuals with a syndrome that carries its own set of social communicative and behavioral difficulties that are not well documented. The aim of this study was to identify the typical range of social communicative impairments observed in children, adolescents, and young adults with DS who do not have comorbid ASD. Methods We examined patterns of scores from the five subscales of the Social Responsiveness Scale (SRS) in 46 individuals with DS (ages 10–21 years) without comorbid ASD relative to the published normative sample. We also explored the correlations between SRS symptomatology and age, nonverbal cognition, and receptive language. Results SRS scores were elevated (i.e., more ASD symptoms endorsed), with mean scores falling into the clinically significant range. Analysis by subscale revealed a specific pattern, with Autistic Mannerisms and Social Cognition scores significantly more elevated than Social Communication scores, which were significantly more elevated than Social Awareness and Social Motivation scores. Correlations between SRS scores and the other measures varied by subscale. Conclusions General elevated ASD symptomatology on the SRS indicates the need for developing population-based norms specific to DS. The pattern of scores across subscales should inform clinicians of the typical range of behaviors observed in DS so that individuals with atypical patterns of behavior can be more easily identified and considered for a full ASD evaluation

Southern Greenland glaciation and Western Boundary Undercurrent evolution recorded on Eirik Drift during the late Pliocene intensification of Northern Hemisphere glaciation

This is the author accepted manuscript. The final version is available from Elsevier via the DOI in this record.We present new sedimentological and environmental magnetic records spanning ~3.2–2.2 Ma, during the intensification of Northern Hemisphere glaciation, from North Atlantic Integrated Ocean Drilling Program Site U1307 on Eirik Drift. Our new datasets and their high-fidelity age control demonstrate that while inland glaciers – and potentially also at times restricted marine-terminating ice-caps – have likely existed on southern Greenland since at least ~3.2 Ma, persistent and extensive marine-terminating glacial margins were only established in this region at 2.72 Ma, ~300 kyr later than in northeastern and eastern Greenland. Despite a dramatic increase in Greenland-sourced ice-rafted debris deposition on Eirik Drift at this time, contemporaneous changes in the bulk magnetic properties of Site U1307 sediments, and a reduction in sediment accumulation rates, suggest a decrease in the delivery of Greenland-sourced glaciofluvial silt, which we attribute to a shift in depositional regime from bottom-current-dominated to glacial-IRD- dominated between ~2.9–2.7 Ma in response to a change in the depth of the flow path of the Western Boundary Undercurrent relative to our study site

An Exactly Conservative Integrator for the n-Body Problem

The two-dimensional n-body problem of classical mechanics is a non-integrable Hamiltonian system for n > 2. Traditional numerical integration algorithms, which are polynomials in the time step, typically lead to systematic drifts in the computed value of the total energy and angular momentum. Even symplectic integration schemes exactly conserve only an approximate Hamiltonian. We present an algorithm that conserves the true Hamiltonian and the total angular momentum to machine precision. It is derived by applying conventional discretizations in a new space obtained by transformation of the dependent variables. We develop the method first for the restricted circular three-body problem, then for the general two-dimensional three-body problem, and finally for the planar n-body problem. Jacobi coordinates are used to reduce the two-dimensional n-body problem to an (n-1)-body problem that incorporates the constant linear momentum and center of mass constraints. For a four-body choreography, we find that a larger time step can be used with our conservative algorithm than with symplectic and conventional integrators.Comment: 17 pages, 3 figures; to appear in J. Phys. A.: Math. Ge

Magnetic unmixing of first-order reversal curve diagrams using principal component analysis

We describe a quantitative magnetic unmixing method based on principal component analysis (PCA) of first-order reversal curve (FORC) diagrams. For PCA we resample FORC distributions on grids that capture diagnostic signatures of single-domain (SD), pseudo-single-domain (PSD), and multi-domain (MD) magnetite, as well as of minerals such as hematite. Individual FORC diagrams are recast as linear combinations of end-member (EM) FORC diagrams, located at user-defined positions in PCA space. The EM selection is guided by constraints derived from physical modeling and imposed by data scatter. We investigate temporal variations of two EMs in bulk North Atlantic sediment cores collected from the Rockall Trough and the Iberian Continental Margin. Sediments from each site contain a mixture of magnetosomes and granulometrically distinct detrital magnetite. We also quantify the spatial variation of three EM components (a coarse silt-sized MD component, a fine silt-sized PSD component, and a mixed clay-sized component containing both SD magnetite and hematite) in surficial sediments along the flow path of the North Atlantic Deep Water (NADW). These samples were separated into granulometric fractions, which helped constrain EM definition. PCA-based unmixing reveals systematic variations in EM relative abundance as a function of distance along NADW flow. Finally, we apply PCA to the combined dataset of Rockall Trough and NADW sediments, which can be recast as a four-EM mixture, providing enhanced discrimination between components. Our method forms the foundation of a general solution to the problem of unmixing multi-component magnetic mixtures, a fundamental task of rock magnetic studies. This article is protected by copyright. All rights reserved

Enhanced ionization in small rare gas clusters

A detailed theoretical investigation of rare gas atom clusters under intense short laser pulses reveals that the mechanism of energy absorption is akin to {\it enhanced ionization} first discovered for diatomic molecules. The phenomenon is robust under changes of the atomic element (neon, argon, krypton, xenon), the number of atoms in the cluster (16 to 30 atoms have been studied) and the fluency of the laser pulse. In contrast to molecules it does not dissappear for circular polarization. We develop an analytical model relating the pulse length for maximum ionization to characteristic parameters of the cluster

A consideration of the challenges involved in supervising international masters students

This paper explores the challenges facing supervisors of international postgraduate students at the dissertation stage of the masters programme. The central problems of time pressure, language difficulties, a lack of critical analysis and a prevalence of personal problems among international students are discussed. This paper makes recommendations for the improvement of language and critical thinking skills, and questions the future policy of language requirements at HE for international Masters students

Use of the Extended Fujita method for representing the molecular weight and molecular weight distributions of native and processed oat beta-glucans

Beta 1–3, 1–4 glucans (“beta-glucans”) are one of the key components of the cell wall of cereals, complementing the main structural component cellulose. Beta-glucans are also an important source of soluble fbre in foods containing oats with claims of other benefcial nutritional properties such as plasma cholesterol lowering in humans. Key to the function of beta-glucans is their molecular weight and because of their high polydispersity - molecular weight distribution. Analytical ultracentrifugation provides a matrix-free approach (not requiring separation columns or media) to polymer molecular weight distribution determination. The sedimentation coefficient distribution is converted to a molecular weight distribution via a power law relation using an established procedure known as the Extended Fujita approach. We establish and apply the power law relation and Extended Fujita method for the frst time to a series of native and processed oat beta-glucans. The application of this approach to beta-glucans from other sources is considered