The Lie-Poisson structure of the reduced n-body problem

The classical n-body problem in d-dimensional space is invariant under the Galilean symmetry group. We reduce by this symmetry group using the method of polynomial invariants. As a result we obtain a reduced system with a Lie-Poisson structure which is isomorphic to sp(2n-2), independently of d. The reduction preserves the natural form of the Hamiltonian as a sum of kinetic energy that depends on velocities only and a potential that depends on positions only. Hence we proceed to construct a Poisson integrator for the reduced n-body problem using a splitting method.Comment: 26 pages, 2 figure

Simultaneous assessment of lung morphology and respiratory motion in retrospectively gated in-vivo microCT of free breathing anesthetized mice

Retrospective gating (RG) is a well established technique in preclinical computed tomography (CT) to assess 3D morphology of the lung. In RG additional angular projections are recorded typically by performing multiple rotations. Consequently, the projections are sorted according to the expansion state of the chest and those sets are then reconstructed separately. Thus, the breathing motion artefacts are suppressed at a cost of strongly elevated X-ray dose levels. Here we propose to use the entire raw data to assess respiratory motion in addition to retrospectively gated 3D reconstruction that visualize anatomical structures of the lung. Using this RG based X-ray respiratory motion measurement approach, which will be referred to as RG based X-ray lung function measurement (rgXLF) on the example of the mdx mouse model of Duchenne muscle dystrophy (mdx) we accurately obtained both the 3D anatomical morphology of the lung and the thoracic bones as well as functional temporal parameters of the lung. Thus, rgXLF will remove the necessity for separate acquisition procedures by being able to reproduce comparable results to the previously established planar X-ray based lung function measurement approach in a single low dose CT scan

Vanishing Twist near Focus-Focus Points

We show that near a focus-focus point in a Liouville integrable Hamiltonian system with two degrees of freedom lines of locally constant rotation number in the image of the energy-momentum map are spirals determined by the eigenvalue of the equilibrium. From this representation of the rotation number we derive that the twist condition for the isoenergetic KAM condition vanishes on a curve in the image of the energy-momentum map that is transversal to the line of constant energy. In contrast to this we also show that the frequency map is non-degenerate for every point in a neighborhood of a focus-focus point.Comment: 13 page

Generic Twistless Bifurcations

We show that in the neighborhood of the tripling bifurcation of a periodic orbit of a Hamiltonian flow or of a fixed point of an area preserving map, there is generically a bifurcation that creates a ``twistless'' torus. At this bifurcation, the twist, which is the derivative of the rotation number with respect to the action, vanishes. The twistless torus moves outward after it is created, and eventually collides with the saddle-center bifurcation that creates the period three orbits. The existence of the twistless bifurcation is responsible for the breakdown of the nondegeneracy condition required in the proof of the KAM theorem for flows or the Moser twist theorem for maps. When the twistless torus has a rational rotation number, there are typically reconnection bifurcations of periodic orbits with that rotation number.Comment: 29 pages, 9 figure

Current Approaches for Image Fusion of Histological Data with Computed Tomography and Magnetic Resonance Imaging

Classical analysis of biological samples requires the destruction of the tissue’s integrity by cutting or grinding it down to thin slices for (Immuno)-histochemical staining and microscopic analysis. Despite high specificity, encoded in the stained 2D section of the whole tissue, the structural information, especially 3D information, is limited. Computed tomography (CT) or magnetic resonance imaging (MRI) scans performed prior to sectioning in combination with image registration algorithms provide an opportunity to regain access to morphological characteristics as well as to relate histological findings to the 3D structure of the local tissue environment. This review provides a summary of prevalent literature addressing the problem of multimodal coregistration of hard- and soft-tissue in microscopy and tomography. Grouped according to the complexity of the dimensions, including image-to-volume (2D ⟶ 3D), image-to-image (2D ⟶ 2D), and volume-to-volume (3D ⟶ 3D), selected currently applied approaches are investigated by comparing the method accuracy with respect to the limiting resolution of the tomography. Correlation of multimodal imaging could position itself as a useful tool allowing for precise histological diagnostic and allow the a priori planning of tissue extraction like biopsies

Maslov Indices and Monodromy

We prove that for a Hamiltonian system on a cotangent bundle that is Liouville-integrable and has monodromy the vector of Maslov indices is an eigenvector of the monodromy matrix with eigenvalue 1. As a corollary the resulting restrictions on the monodromy matrix are derived.Comment: 6 page

Non-Invasive Optical Motion Tracking Allows Monitoring of Respiratory Dynamics in Dystrophin-Deficient Mice

Duchenne muscular dystrophy (DMD) is the most common x-chromosomal inherited dystrophinopathy which leads to progressive muscle weakness and a premature death due to cardiorespiratory dysfunction. The mdx mouse lacks functional dystrophin protein and has a comparatively human-like diaphragm phenotype. To date, diaphragm function can only be inadequately mapped in preclinical studies and a simple reliable translatable method of tracking the severity of the disease still lacks. We aimed to establish a sensitive, reliable, harmless and easy way to assess the effects of respiratory muscle weakness and subsequent irregularity in breathing pattern. Optical respiratory dynamics tracking (ORDT) was developed utilising a camera to track the movement of paper markers placed on the thoracic-abdominal region of the mouse. ORDT successfully distinguished diseased mdx phenotype from healthy controls by measuring significantly higher expiration constants (k) in mdx mice compared to wildtype (wt), which were also observed in the established X-ray based lung function (XLF). In contrast to XLF, with ORDT we were able to distinguish distinct fast and slow expiratory phases. In mdx mice, a larger part of the expiratory marker displacement was achieved in this initial fast phase as compared to wt mice. This phenomenon could not be observed in the XLF measurements. We further validated the simplicity and reliability of our approach by demonstrating that it can be performed using free-hand smartphone acquisition. We conclude that ORDT has a great preclinical potential to monitor DMD and other neuromuscular diseases based on changes in the breathing patterns with the future possibility to track therapy response