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

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

A maximum density rule for surfaces of quasicrystals

A rule due to Bravais of wide validity for crystals is that their surfaces correspond to the densest planes of atoms in the bulk of the material. Comparing a theoretical model of i-AlPdMn with experimental results, we find that this correspondence breaks down and that surfaces parallel to the densest planes in the bulk are not the most stable, i.e. they are not so-called bulk terminations. The correspondence can be restored by recognizing that there is a contribution to the surface not just from one geometrical plane but from a layer of stacked atoms, possibly containing more than one plane. We find that not only does the stability of high-symmetry surfaces match the density of the corresponding layer-like bulk terminations but the exact spacings between surface terraces and their degree of pittedness may be determined by a simple analysis of the density of layers predicted by the bulk geometric model.Comment: 8 pages of ps-file, 3 Figs (jpg

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

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

Track billiards

We study a class of planar billiards having the remarkable property that their phase space consists up to a set of zero measure of two invariant sets formed by orbits moving in opposite directions. The tables of these billiards are tubular neighborhoods of differentiable Jordan curves that are unions of finitely many segments and arcs of circles. We prove that under proper conditions on the segments and the arcs, the billiards considered have non-zero Lyapunov exponents almost everywhere. These results are then extended to a similar class of of 3-dimensional billiards. Finally, we find that for some subclasses of track billiards, the mechanism generating hyperbolicity is not the defocusing one that requires every infinitesimal beam of parallel rays to defocus after every reflection off of the focusing boundary.Comment: 7 figure

Oxidation and magnetic states of chalcopyrite CuFeS2: a first principles calculation

The ground state band structure, magnetic moments, charges and population numbers of electronic shells of Cu and Fe atoms have been calculated for chalcopyrite CuFeS2 using density functional theory. The comparison between our calculation results and experimental data (X ray photoemission, X ray absorption and neutron diffraction spectroscopy) has been made. Our calculations predict a formal oxidation state for chalcopyrite as Cu1+Fe3+S. However, the assignment of formal valence state to transition metal atoms appears to be oversimplified. It is anticipated that the valence state can be confirmed experimentally by nuclear magnetic and nuclear quadrupole resonance and Mössbauer spectroscopy methods

Highly efficient separation of actinides from lanthanides by a phenanthroline-derived bis-triazine ligand

The synthesis, lanthanide complexation, and solvent ex- traction of actinide(III) and lanthanide(III) radiotracers from nitric acid solutions by a phenanthroline-derived quadridentate bis-triazine ligand are described. The ligand separates Am(III) and Cm(III) from the lanthanides with remarkably high efficiency, high selectivity, and fast extraction kinetics compared to its 2,2'-bipyridine counterpart. Structures of the 1:2 bis-complexes of the ligand with Eu(III) and Yb(III) were elucidated by X-ray crystallography and force field calculations, respec-tively. The Eu(III) bis-complex is the first 1:2 bis-complex of a quadridentate bis-triazine ligand to be characterized by crystallography. The faster rates of extraction were verified by kinetics measurements using the rotating membrane cell technique in several diluents. The improved kinetics of metal ion extraction are related to the higher surface activity of the ligand at the phase interface. The improvement in the ligand's properties on replacing the bipyridine unit with a phenanthroline unit far exceeds what was anticipated based on ligand design alone