Approximating multi-dimensional Hamiltonian flows by billiards

Consider a family of smooth potentials $V_{\epsilon}$, which, in the limit $\epsilon\to0$, become a singular hard-wall potential of a multi-dimensional billiard. We define auxiliary billiard domains that asymptote, as $\epsilon\to0$ to the original billiard, and provide asymptotic expansion of the smooth Hamiltonian solution in terms of these billiard approximations. The asymptotic expansion includes error estimates in the $C^{r}$ norm and an iteration scheme for improving this approximation. Applying this theory to smooth potentials which limit to the multi-dimensional close to ellipsoidal billiards, we predict when the separatrix splitting persists for various types of potentials

Super resolution using sparse sampling at portable ultra-low field MR

Ultra-low field (ULF) magnetic resonance imaging (MRI) holds the potential to make MRI more accessible, given its cost-effectiveness, reduced power requirements, and portability. However, signal-to-noise ratio (SNR) drops with field strength, necessitating imaging with lower resolution and longer scan times. This study introduces a novel Fourier-based Super Resolution (FouSR) approach, designed to enhance the resolution of ULF MRI images with minimal increase in total scan time. FouSR combines spatial frequencies from two orthogonal ULF images of anisotropic resolution to create an isotropic T2-weighted fluid-attenuated inversion recovery (FLAIR) image. We hypothesized that FouSR could effectively recover information from under-sampled slice directions, thereby improving the delineation of multiple sclerosis (MS) lesions and other significant anatomical features. Importantly, the FouSR algorithm can be implemented on the scanner with changes to the k-space trajectory. Paired ULF (Hyperfine SWOOP, 0.064 tesla) and high field (Siemens, Skyra, 3 Tesla) FLAIR scans were collected on the same day from a phantom and a cohort of 10 participants with MS or suspected MS (6 female; mean ± SD age: 44.1 ± 4.1). ULF scans were acquired along both coronal and axial planes, featuring an in-plane resolution of 1.7 mm × 1.7 mm with a slice thickness of 5 mm. FouSR was evaluated against registered ULF coronal and axial scans, their average (ULF average) and a gold standard SR (ANTs SR). FouSR exhibited higher SNR (47.96 ± 12.6) compared to ULF coronal (36.7 ± 12.2) and higher lesion conspicuity (0.12 ± 0.06) compared to ULF axial (0.13 ± 0.07) but did not exhibit any significant differences contrast-to-noise-ratio (CNR) compared to other methods in patient scans. However, FouSR demonstrated superior image sharpness (0.025 ± 0.0040) compared to all other techniques (ULF coronal 0.021 ± 0.0037, q = 5.9, p-adj. = 0.011; ULF axial 0.018 ± 0.0026, q = 11.1, p-adj. = 0.0001; ULF average 0.019 ± 0.0034, q = 24.2, p-adj. < 0.0001) and higher lesion sharpness (−0.97 ± 0.31) when compared to the ULF average (−1.02 ± 0.37, t(543) = −10.174, p = <0.0001). Average blinded qualitative assessment by three experienced MS neurologists showed no significant difference in WML and sulci or gyri visualization between FouSR and other methods. FouSR can, in principle, be implemented on the scanner to produce clinically useful FLAIR images at higher resolution on the fly, providing a valuable tool for visualizing lesions and other anatomical structures in MS

Billiards with polynomial mixing rates

While many dynamical systems of mechanical origin, in particular billiards, are strongly chaotic -- enjoy exponential mixing, the rates of mixing in many other models are slow (algebraic, or polynomial). The dynamics in the latter are intermittent between regular and chaotic, which makes them particularly interesting in physical studies. However, mathematical methods for the analysis of systems with slow mixing rates were developed just recently and are still difficult to apply to realistic models. Here we reduce those methods to a practical scheme that allows us to obtain a nearly optimal bound on mixing rates. We demonstrate how the method works by applying it to several classes of chaotic billiards with slow mixing as well as discuss a few examples where the method, in its present form, fails.Comment: 39pages, 11 figue

Convergence of invariant densities in the small-noise limit

This paper presents a systematic numerical study of the effects of noise on the invariant probability densities of dynamical systems with varying degrees of hyperbolicity. It is found that the rate of convergence of invariant densities in the small-noise limit is frequently governed by power laws. In addition, a simple heuristic is proposed and found to correctly predict the power law exponent in exponentially mixing systems. In systems which are not exponentially mixing, the heuristic provides only an upper bound on the power law exponent. As this numerical study requires the computation of invariant densities across more than 2 decades of noise amplitudes, it also provides an opportunity to discuss and compare standard numerical methods for computing invariant probability densities.Comment: 27 pages, 19 figures, revised with minor correction

On stochastic sea of the standard map

Consider a generic one-parameter unfolding of a homoclinic tangency of an area preserving surface diffeomorphism. We show that for many parameters (residual subset in an open set approaching the critical value) the corresponding diffeomorphism has a transitive invariant set $\Omega$ of full Hausdorff dimension. The set $\Omega$ is a topological limit of hyperbolic sets and is accumulated by elliptic islands. As an application we prove that stochastic sea of the standard map has full Hausdorff dimension for sufficiently large topologically generic parameters.Comment: 36 pages, 5 figure

Growth and dislocation studies of β-HMX

Background: The defect structure of organic materials is important as it plays a major role in their crystal growth properties. It also can play a subcritical role in “hot-spot” detonation processes of energetics and one such energetic is cyclotetramethylene-tetranitramine, in the commonly used beta form (β-HMX). Results: The as-grown crystals grown by evaporation from acetone show prismatic, tabular and columnar habits, all with {011}, {110}, (010) and (101) faces. Etching on (010) surfaces revealed three different types of etch pits, two of which could be identified with either pure screw or pure edge dislocations, the third is shown to be an artifact of the twinning process that this material undergoes. Examination of the {011} and {110} surfaces show only one type of etch pit on each surface; however their natural asymmetry precludes the easy identification of their Burgers vector or dislocation type. Etching of cleaved {011} surfaces demonstrates that the etch pits can be associated with line dislocations. All dislocations appear randomly on the crystal surfaces and do not form alignments characteristic of mechanical deformation by dislocation slip. Conclusions: Crystals of β-HMX grown from acetone show good morphological agreement with that predicted by modelling, with three distinct crystal habits observed depending upon the supersaturation of the growth solution. Prismatic habit was favoured at low supersaturation, while tabular and columnar crystals were predominant at higher super saturations. The twin plane in β-HMX was identified as a (101) reflection plane. The low plasticity of β-HMX is shown by the lack of etch pit alignments corresponding to mechanically induced dislocation arrays. On untwinned {010} faces, two types of dislocations exist, pure edge dislocations with b = [010] and pure screw dislocations with b = [010]. On twinned (010) faces, a third dislocation type exists and it is proposed that these pits are associated with pure screw dislocations with b = [010]

How metal films de-wet substrates - identifying the kinetic pathways and energetic driving forces

We study how single-crystal chromium films of uniform thickness on W(110) substrates are converted to arrays of three-dimensional (3D) Cr islands during annealing. We use low-energy electron microscopy (LEEM) to directly observe a kinetic pathway that produces trenches that expose the wetting layer. Adjacent film steps move simultaneously uphill and downhill relative to the staircase of atomic steps on the substrate. This step motion thickens the film regions where steps advance. Where film steps retract, the film thins, eventually exposing the stable wetting layer. Since our analysis shows that thick Cr films have a lattice constant close to bulk Cr, we propose that surface and interface stress provide a possible driving force for the observed morphological instability. Atomistic simulations and analytic elastic models show that surface and interface stress can cause a dependence of film energy on thickness that leads to an instability to simultaneous thinning and thickening. We observe that de-wetting is also initiated at bunches of substrate steps in two other systems, Ag/W(110) and Ag/Ru(0001). We additionally describe how Cr films are converted into patterns of unidirectional stripes as the trenches that expose the wetting layer lengthen along the W[001] direction. Finally, we observe how 3D Cr islands form directly during film growth at elevated temperature. The Cr mesas (wedges) form as Cr film steps advance down the staircase of substrate steps, another example of the critical role that substrate steps play in 3D island formation