Counting Berg partitions

We call a Markov partition of a two dimensional hyperbolic toral automorphism a Berg partition if it contains just two rectangles. We describe all Berg partitions for a given hyperbolic toral automorphism. In particular there are exactly (k + n + l + m)/2 nonequivalent Berg partitions with the same connectivity matrix (k, l, m, n)

Classification of biological micro-objects using optical coherence tomography: in silico study

We report on the development of a technique for differentiating between biological micro-objects using a rigorous, full-wave model of OCT image formation. We model an existing experimental prototype which uses OCT to interrogate a microfluidic chip containing the blood cells. A full-wave model is required since the technique uses light back-scattered by a scattering substrate, rather than by the cells directly. The light back-scattered by the substrate is perturbed upon propagation through the cells, which flow between the substrate and imaging system’s objective lens. We present the key elements of the 3D, Maxwell equation-based computational model, the key findings of the computational study and a comparison with experimental results

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

Linear stability in billiards with potential

A general formula for the linearized Poincar\'e map of a billiard with a potential is derived. The stability of periodic orbits is given by the trace of a product of matrices describing the piecewise free motion between reflections and the contributions from the reflections alone. For the case without potential this gives well known formulas. Four billiards with potentials for which the free motion is integrable are treated as examples: The linear gravitational potential, the constant magnetic field, the harmonic potential, and a billiard in a rotating frame of reference, imitating the restricted three body problem. The linear stability of periodic orbits with period one and two is analyzed with the help of stability diagrams, showing the essential parameter dependence of the residue of the periodic orbits for these examples.Comment: 22 pages, LaTex, 4 Figure

Optical Coherence Tomography (OCT) for examination of artworks

Chapter in the book: Bastidas D., Cano E. (eds) Advanced Characterization Techniques, Diagnostic Tools and Evaluation Methods in Heritage Science. Springer, Cham, 2018, pp 49-59 , doi: 10.1007/978-3-319-75316-4, Authors’version after embargo periodOptical coherence tomography is a fast, non-invasive technique of structural analysis utilising near-infrared radiation. Examples of using OCT, for obtaining cross-sectional images of objects of craftsmanship and an easel painting have been shown. Issues regarding the technique of execution and destruction phenomena were resolved non-invasively. In some cases, the secondary alterations can be identified and localised within the object’s structure which helps in authentication of the artwork

Recurrence and higher ergodic properties for quenched random Lorentz tubes in dimension bigger than two

We consider the billiard dynamics in a non-compact set of R^d that is constructed as a bi-infinite chain of translated copies of the same d-dimensional polytope. A random configuration of semi-dispersing scatterers is placed in each copy. The ensemble of dynamical systems thus defined, one for each global realization of the scatterers, is called `quenched random Lorentz tube'. Under some fairly general conditions, we prove that every system in the ensemble is hyperbolic and almost every system is recurrent, ergodic, and enjoys some higher chaotic properties.Comment: Final version for J. Stat. Phys., 18 pages, 4 figure

Optimum spectral window for imaging of art with optical coherence tomography

Optical Coherence Tomography (OCT) has been shown to have potential for important applications in the field of art conservation and archaeology due to its ability to image subsurface microstructures non-invasively. However, its depth of penetration in painted objects is limited due to the strong scattering properties of artists’ paints. VIS-NIR (400 nm – 2400 nm) reflectance spectra of a wide variety of paints made with historic artists’ pigments have been measured. The best spectral window with which to use optical coherence tomography (OCT) for the imaging of subsurface structure of paintings was found to be around 2.2 μm. The same spectral window would also be most suitable for direct infrared imaging of preparatory sketches under the paint layers. The reflectance spectra from a large sample of chemically verified pigments provide information on the spectral transparency of historic artists’ pigments/paints as well as a reference set of spectra for pigment identification. The results of the paper suggest that broadband sources at ~2 microns are highly desirable for OCT applications in art and potentially material science in general

Spectral Statistics in the Quantized Cardioid Billiard

The spectral statistics in the strongly chaotic cardioid billiard are studied. The analysis is based on the first 11000 quantal energy levels for odd and even symmetry respectively. It is found that the level-spacing distribution is in good agreement with the GOE distribution of random-matrix theory. In case of the number variance and rigidity we observe agreement with the random-matrix model for short-range correlations only, whereas for long-range correlations both statistics saturate in agreement with semiclassical expectations. Furthermore the conjecture that for classically chaotic systems the normalized mode fluctuations have a universal Gaussian distribution with unit variance is tested and found to be in very good agreement for both symmetry classes. By means of the Gutzwiller trace formula the trace of the cosine-modulated heat kernel is studied. Since the billiard boundary is focusing there are conjugate points giving rise to zeros at the locations of the periodic orbits instead of exclusively Gaussian peaks.Comment: 20 pages, uu-encoded ps.Z-fil

About ergodicity in the family of limacon billiards

By continuation from the hyperbolic limit of the cardioid billiard we show that there is an abundance of bifurcations in the family of limacon billiards. The statistics of these bifurcation shows that the size of the stable intervals decreases with approximately the same rate as their number increases with the period. In particular, we give numerical evidence that arbitrarily close to the cardioid there are elliptic islands due to orbits created in saddle node bifurcations. This shows explicitly that if in this one parameter family of maps ergodicity occurs for more than one parameter the set of these parameter values has a complicated structure.Comment: 17 pages, 9 figure