Correlations for pairs of periodic trajectories for open billiards

In this paper we prove two asymptotic estimates for pairs of closed trajectories for open billiards similar to those established by Pollicott and Sharp for closed geodesics on negatively curved compact surfaces. The first of these estimates holds for general open billiards in any dimension. The more intricate second estimate is established for open billiards satisfying the so called Dolgopyat type estimates. This class of billiards includes all open billiards in the plane and open billiards in $\R^N, N \geq 3$ satisfying some additional conditions

Role of Landau-Rabi quantization of electron motion on the crust of magnetars within the nuclear energy density functional theory

Magnetic fields of order $10^{15}$ G have been measured at the surface of some neutron stars, and much stronger magnetic fields are expected to be present in the solid region beneath the surface. The effects of the magnetic field on the equation of state and on the composition of the crust due to Landau-Rabi quantization of electron motion are studied. Both the outer and inner crustal regions are described in a unified and consistent way within the nuclear-energy density functional theory.Comment: 23 pages, 11 figure

Landau quantization and neutron emissions by nuclei in the crust of a magnetar

Magnetars are neutron stars endowed with surface magnetic fields of the order of $10^{14}-10^{15}$~G, and with presumably much stronger fields in their interior. As a result of Landau quantization of electron motion, the neutron-drip transition in the crust of a magnetar is shifted to either higher or lower densities depending on the magnetic field strength. The impact of nuclear uncertainties is explored considering the recent series of Brussels-Montreal microscopic nuclear mass models. All these models are based on the Hartree-Fock-Bogoliubov method with generalized Skyrme functionals. They differ in their predictions for the symmetry energy coefficient at saturation, and for the stiffness of the neutron-matter equation of state. For comparison, we have also considered the very accurate but more phenomenological model of Duflo and Zuker. Although the equilibrium composition of the crust of a magnetar and the onset of neutron emission are found to be model dependent, the quantum oscillations of the threshold density are essentially universal.Comment: 7 pages, 2 figure

FedDP: Dual Personalization in Federated Medical Image Segmentation

Personalized federated learning (PFL) addresses the data heterogeneity challenge faced by general federated learning (GFL). Rather than learning a single global model, with PFL a collection of models are adapted to the unique feature distribution of each site. However, current PFL methods rarely consider self-attention networks which can handle data heterogeneity by long-range dependency modeling and they do not utilize prediction inconsistencies in local models as an indicator of site uniqueness. In this paper, we propose FedDP, a novel federated learning scheme with dual personalization, which improves model personalization from both feature and prediction aspects to boost image segmentation results. We leverage long-range dependencies by designing a local query (LQ) that decouples the query embedding layer out of each local model, whose parameters are trained privately to better adapt to the respective feature distribution of the site. We then propose inconsistency-guided calibration (IGC), which exploits the inter-site prediction inconsistencies to accommodate the model learning concentration. By encouraging a model to penalize pixels with larger inconsistencies, we better tailor prediction-level patterns to each local site. Experimentally, we compare FedDP with the state-of-the-art PFL methods on two popular medical image segmentation tasks with different modalities, where our results consistently outperform others on both tasks. Our code and models will be available at https://github.com/jcwang123/PFL-Seg-Trans

A Simulation Study of Spectral Cerenkov Luminescence Imaging for Tumour Margin Estimation

Breast cancer is the most common cancer in women in the world. Breast-conserving surgery (BCS) is a standard surgical treatment for breast cancer with the key objective of removing breast tissue, maintaining a negative surgical margin and providing a good cosmetic outcome. A positive surgical margin, meaning the presence of cancerous tissues on the surface of the breast specimen after surgery, is associated with local recurrence after therapy. In this study, we investigate a new imaging modality based on Cerenkov luminescence imaging (CLI) for the purpose of detecting positive surgical margins during BCS. We develop Monte Carlo (MC) simulations using the Geant4 nuclear physics simulation toolbox to study the spectrum of photons emitted given 18F-FDG and breast tissue properties. The resulting simulation spectra show that the CLI signal contains information that may be used to estimate whether the cancerous cells are at a depth of less than 1 mm or greater than 1 mm given appropriate imaging system design and sensitivity. The simulation spectra also show that when the source is located within 1 mm of the surface, the tissue parameters are not relevant to the model as the spectra do not vary significantly. At larger depths, however, the spectral information varies significantly with breast optical parameters, having implications for further studies and system design. While promising, further studies are needed to quantify the CLI response to more accurately incorporate tissue specific parameters and patient specific anatomical details