Periodic orbit sum rules for billiards: Accelerating cycle expansions

We show that the periodic orbit sums for 2-dimensional billiards satisfy an infinity of exact sum rules. We test such sum rules and demonstrate that they can be used to accelerate the convergence of cycle expansions for averages such as Lyapunov exponents.Comment: 19 pages, 5 postscript figures, submitted to Journal of Physics

Periodic orbit quantization of the Sinai billiard in the small scatterer limit

We consider the semiclassical quantization of the Sinai billiard for disk radii R small compared to the wave length 2 pi/k. Via the application of the periodic orbit theory of diffraction we derive the semiclassical spectral determinant. The limitations of the derived determinant are studied by comparing it to the exact KKR determinant, which we generalize here for the A_1 subspace. With the help of the Ewald resummation method developed for the full KKR determinant we transfer the complex diffractive determinant to a real form. The real zeros of the determinant are the quantum eigenvalues in semiclassical approximation. The essential parameter is the strength of the scatterer c=J_0(kR)/Y_0(kR). Surprisingly, this can take any value between plus and minus infinity within the range of validity of the diffractive approximation kR <<4. We study the statistics exhibited by spectra for fixed values of c. It is Poissonian for |c|=infinity, provided the disk is placed inside a rectangle whose sides obeys some constraints. For c=0 we find a good agreement of the level spacing distribution with GOE, whereas the form factor and two-point correlation function are similar but exhibit larger deviations. By varying the parameter c from 0 to infinity the level statistics interpolates smoothly between these limiting cases.Comment: 17 pages LaTeX, 5 postscript figures, submitted to J. Phys. A: Math. Ge

Deterministic stream-sampling for probabilistic programming: semantics and verification

Probabilistic programming languages rely fundamentally on some notion of sampling, and this is doubly true for probabilistic programming languages which perform Bayesian inference using Monte Carlo techniques. Verifying samplers - proving that they generate samples from the correct distribution - is crucial to the use of probabilistic programming languages for statistical modelling and inference. However, the typical denotational semantics of probabilistic programs is incompatible with deterministic notions of sampling. This is problematic, considering that most statistical inference is performed using pseudorandom number generators.We present a higher-order probabilistic programming language centred on the notion of samplers and sampler operations. We give this language an operational and denotational semantics in terms of continuous maps between topological spaces. Our language also supports discontinuous operations, such as comparisons between reals, by using the type system to track discontinuities. This feature might be of independent interest, for example in the context of differentiable programming.Using this language, we develop tools for the formal verification of sampler correctness. We present an equational calculus to reason about equivalence of samplers, and a sound calculus to prove semantic correctness of samplers, i.e. that a sampler correctly targets a given measure by construction

On the duality between periodic orbit statistics and quantum level statistics

We discuss consequences of a recent observation that the sequence of periodic orbits in a chaotic billiard behaves like a poissonian stochastic process on small scales. This enables the semiclassical form factor $K_{sc}(\tau)$ to agree with predictions of random matrix theories for other than infinitesimal $\tau$ in the semiclassical limit.Comment: 8 pages LaTe

Auto-RSM: An automated parameter-selection algorithm for the RSM map exoplanet detection algorithm

Context. Most of the high-contrast imaging (HCI) data-processing techniques used over the last 15 years have relied on the angular differential imaging (ADI) observing strategy, along with subtraction of a reference point spread function (PSF) to generate exoplanet detection maps. Recently, a new algorithm called regime switching model (RSM) map has been proposed to take advantage of these numerous PSF-subtraction techniques; RSM uses several of these techniques to generate a single probability map. Selection of the optimal parameters for these PSF-subtraction techniques as well as for the RSM map is not straightforward, is time consuming, and can be biased by assumptions made as to the underlying data set. Aims: We propose a novel optimisation procedure that can be applied to each of the PSF-subtraction techniques alone, or to the entire RSM framework. Methods: The optimisation procedure consists of three main steps: (i) definition of the optimal set of parameters for the PSF-subtraction techniques using the contrast as performance metric, (ii) optimisation of the RSM algorithm, and (iii) selection of the optimal set of PSF-subtraction techniques and ADI sequences used to generate the final RSM probability map. Results: The optimisation procedure is applied to the data sets of the exoplanet imaging data challenge, which provides tools to compare the performance of HCI data-processing techniques. The data sets consist of ADI sequences obtained with three state-of-the-art HCI instruments: SPHERE, NIRC2, and LMIRCam. The results of our analysis demonstrate the interest of the proposed optimisation procedure, with better performance metrics compared to the earlier version of RSM, as well as to other HCI data-processing techniques.EPIC; NNEx

Cross-talk between the Notch and TGF-β signaling pathways mediated by interaction of the Notch intracellular domain with Smad3

The Notch and transforming growth factor-β (TGF-β) signaling pathways play critical roles in the control of cell fate during metazoan development. However, mechanisms of cross-talk and signal integration between the two systems are unknown. Here, we demonstrate a functional synergism between Notch and TGF-β signaling in the regulation of Hes-1, a direct target of the Notch pathway. Activation of TGF-β signaling up-regulated Hes-1 expression in vitro and in vivo. This effect was abrogated in myogenic cells by a dominant-negative form of CSL, an essential DNA-binding component of the Notch pathway. TGF-β regulated transcription from the Hes-1 promoter in a Notch-dependent manner, and the intracellular domain of Notch1 (NICD) cooperated synergistically with Smad3, an intracellular transducer of TGF-β signals, to induce the activation of synthetic promoters containing multimerized CSL- or Smad3-binding sites. NICD and Smad3 were shown to interact directly, both in vitro and in cells, in a ligand-dependent manner, and Smad3 could be recruited to CSL-binding sites on DNA in the presence of CSL and NICD. These findings indicate that Notch and TGF-β signals are integrated by direct protein–protein interactions between the signal-transducing intracellular elements from both pathways

Recent Results on the Periodic Lorentz Gas

The Drude-Lorentz model for the motion of electrons in a solid is a classical model in statistical mechanics, where electrons are represented as point particles bouncing on a fixed system of obstacles (the atoms in the solid). Under some appropriate scaling assumption -- known as the Boltzmann-Grad scaling by analogy with the kinetic theory of rarefied gases -- this system can be described in some limit by a linear Boltzmann equation, assuming that the configuration of obstacles is random [G. Gallavotti, [Phys. Rev. (2) vol. 185 (1969), 308]). The case of a periodic configuration of obstacles (like atoms in a crystal) leads to a completely different limiting dynamics. These lecture notes review several results on this problem obtained in the past decade as joint work with J. Bourgain, E. Caglioti and B. Wennberg.Comment: 62 pages. Course at the conference "Topics in PDEs and applications 2008" held in Granada, April 7-11 2008; figure 13 and a misprint in Theorem 4.6 corrected in the new versio

Negative length orbits in normal-superconductor billiard systems

The Path-Length Spectra of mesoscopic systems including diffractive scatterers and connected to superconductor is studied theoretically. We show that the spectra differs fundamentally from that of normal systems due to the presence of Andreev reflection. It is shown that negative path-lengths should arise in the spectra as opposed to normal system. To highlight this effect we carried out both quantum mechanical and semiclassical calculations for the simplest possible diffractive scatterer. The most pronounced peaks in the Path-Length Spectra of the reflection amplitude are identified by the routes that the electron and/or hole travels.Comment: 4 pages, 4 figures include