Machine Learning meets Number Theory: The Data Science of Birch-Swinnerton-Dyer

Empirical analysis is often the first step towards the birth of a conjecture. This is the case of the Birch-Swinnerton-Dyer (BSD) Conjecture describing the rational points on an elliptic curve, one of the most celebrated unsolved problems in mathematics. Here we extend the original empirical approach, to the analysis of the Cremona database of quantities relevant to BSD, inspecting more than 2.5 million elliptic curves by means of the latest techniques in data science, machine-learning and topological data analysis. Key quantities such as rank, Weierstrass coefficients, period, conductor, Tamagawa number, regulator and order of the Tate-Shafarevich group give rise to a high-dimensional point-cloud whose statistical properties we investigate. We reveal patterns and distributions in the rank versus Weierstrass coefficients, as well as the Beta distribution of the BSD ratio of the quantities. Via gradient boosted trees, machine learning is applied in finding inter-correlation amongst the various quantities. We anticipate that our approach will spark further research on the statistical properties of large datasets in Number Theory and more in general in pure Mathematics

Thermodynamic analysis of a novel fossil-fuel–free energy storage system with a trans-critical carbon dioxide cycle and heat pump

This paper presents and analyzes a novel fossil-fuel–free trans-critical energy storage system that uses CO2 as the working fluid in a closed loop shuttled between two saline aquifers or caverns at different depths: one a low-pressure reservoir and the other a high-pressure reservoir. Thermal energy storage and a heat pump are adopted to eliminate the need for external natural gas for heating the CO2 entering the energy recovery turbines. We carefully analyze the energy storage and recovery processes to reveal the actual efficiency of the system. We also highlight thermodynamic and sensitivity analyses of the performance of this fossil-fuel–free trans-critical energy storage system based on a steady-state mathematical method. It is found that the fossil-fuel–free trans-critical CO2 energy storage system has good comprehensive thermodynamic performance. The exergy efficiency, round-trip efficiency, and energy storage efficiency are 67.89%, 66%, and 58.41%, and the energy generated of per unit storage volume is 2.12 kW·h/m3, and the main contribution to exergy destruction is the turbine reheater, from which we can quantify how performance can be improved. Moreover, with a higher energy storage and recovery pressure and lower pressure in the low-pressure reservoir, this novel system shows promising performance

Highly Stable Luminous "snakes" from CsPbX3 Perovskite Nanocrystals Anchored on Amine-Coated Silica Nanowires

CsPbX3 (X = Cl, Br, and I) perovskite nanocrystals (NCs) are known for their exceptional optoelectronic properties, yet the material's instability toward polar solvents, heat, or UV irradiation greatly limits its further applications. Herein, an efficient in situ growing strategy has been developed to give highly stable perovskite NC composites (abbreviated CsPbX3@CA-SiO2) by anchoring CsPbX3 NCs onto silica nanowires (NWs), which effectively depresses the optical degradation of their photoluminescence (PL) and enhances stability. The preparation of surface-functionalized serpentine silica NWs is realized by a sol-gel process involving hydrolysis of a mixture of tetraethyl orthosilicate (TEOS), 3-aminopropyltriethoxysilane (APTES), and trimethoxy(octadecyl)silane (TMODS) in a water/oil emulsion. The serpentine NWs are formed via an anisotropic growth with lengths up to 8 μm. The free amino groups are employed as surface ligands for growing perovskite NCs, yielding distributed monodisperse NCs (∼8 nm) around the NW matrix. The emission wavelength is tunable by simple variation of the halide compositions (CsPbX3, X = Cl, Br, or I), and the composites demonstrate a high photoluminescence quantum yield (PLQY 32-69%). Additionally, we have demonstrated the composites CsPbX3@CA-SiO2 can be self-woven to form a porous 3D hierarchical NWs membrane, giving rise to a superhydrophobic surface with hierarchical micro/nano structural features. The resulting composites exhibit high stability toward water, heat, and UV irradiation. This work elucidates an effective strategy to incorporate perovskite nanocrystals onto functional matrices as multifunctional stable light sources

Radiance and Doppler shift distributions across the network of the quiet Sun

The radiance and Doppler-shift distributions across the solar network provide observational constraints of two-dimensional modeling of transition-region emission and flows in coronal funnels. Two different methods, dispersion plots and average-profile studies, were applied to investigate these distributions. In the dispersion plots, we divided the entire scanned region into a bright and a dark part according to an image of Fe xii; we plotted intensities and Doppler shifts in each bin as determined according to a filtered intensity of Si ii. We also studied the difference in height variations of the magnetic field as extrapolated from the MDI magnetogram, in and outside network. For the average-profile study, we selected 74 individual cases and derived the average profiles of intensities and Doppler shifts across the network. The dispersion plots reveal that the intensities of Si ii and C iv increase from network boundary to network center in both parts. However, the intensity of Ne viii shows different trends, namely increasing in the bright part and decreasing in the dark part. In both parts, the Doppler shift of C iv increases steadily from internetwork to network center. The average-profile study reveals that the intensities of the three lines all decline from the network center to internetwork region. The binned intensities of Si ii and Ne viii have a good correlation. We also find that the large blue shift of Ne viii does not coincide with large red shift of C iv. Our results suggest that the network structure is still prominent at the layer where Ne viii is formed in the quiet Sun, and that the magnetic structures expand more strongly in the dark part than in the bright part of this quiet Sun region.Comment: 10 pages,9 figure