Learning rate is arguably the most important hyper-parameter to tune when training a neural network. As manually setting right learning rate remains a cumbersome process, adaptive learning rate algorithms aim at automating such a process. Motivated by the success of the Barzilai–Borwein (BB) step-size method in many gradient descent methods for solving convex problems, this paper aims at investigating the potential of the BB method for training neural networks. With strong motivation from related convergence analysis, the BB method is generalized to adaptive learning rate of mini-batch gradient descent. The experiments showed that, in contrast to many existing methods, the proposed BB method is highly insensitive to initial learning rate, especially in terms of generalization performance. Also, the BB method showed its advantages on both learning speed and generalization performance over other available methods.
We derive the definition of the Berry phase for adiabatic transport of a composite Fermion (CF) in a half-filled composite Fermi-liquid (CFL). It is found to be different from that adopted in previous investigations by Geraedts et al. With the definition, the numerical evaluation of the Berry phase becomes robust and free of extraneous phase factors. We show that the two forms of microscopic wave-functions of the CFL, i.e., the Jain-Kamilla type wave function and the standard CF wave function, yield different distributions of the Berry curvature in the momentum space. For the former, the Berry curvature has a continuous distribution inside the Fermi sea and vanishes outside, whereas for the latter, the Berry curvature is uniform in the whole momentum space. To facilitate an analytic derivation for the latter, we reveal a simple structure of standard CF wave functions by establishing their connections to the Segal-Bargmann transform. We conclude that the CF with respect to both the microscopic wave-functions is not a massless Dirac particle.
Abstract Identifying hotspots of species richness/rarity is the most commonly used approach worldwide for defining areas of high conservation importance. However, the use of the hotspot method limits one's ability to protect or exclude particular species as all species are treated equally. Particularly, range-restricted species require high conservation attention because they are more vulnerable than common species. However, the efficiency of the hotspot method in capturing range-restricted species is yet to be explored, although it is known that this method provides low species coverage. Here, using a comprehensive database of Chinese woody plants, we mapped the diversity pattern of 11,405 species at the spatial resolution of 50 × 50 km2 and identified hotspot areas using 1, 2.5, 5, 10, 25, and 50% thresholds. We then evaluated the proportion of range-restricted versus common species captured/missed by each hotspot threshold. We found that the commonly used hotspot thresholds (5 and 10%) failed to capture 41–45% of range-restricted species, which indicates that using the hotspot method for conservation prioritization exposes range-restricted species to high extinction risk. Relying entirely on the hotspot method to prioritize conservation areas, therefore, can be risky not only because it provides low species coverage but also because the missed species are mostly range-restricted species. We advocate adopting more efficient methods, such as systematic conservation planning, rather than the hotspot method, to increase the coverage of range-restricted species in designated priority areas and balance the needs of biodiversity conservation and economic development.
A branching random walk algorithm for many-body Wigner equations and its numerical applications for quantum dynamics in phase space are proposed and analyzed in this paper. Using an auxiliary function, the truncated Wigner equation and its adjoint form are cast into integral formulations, which can be then reformulated into renewal-type equations with probabilistic interpretations. We prove that the first moment of a branching random walk is the solution for the adjoint equation. With the help of the additional degree of freedom offered by the auxiliary function, we are able to produce a weighted-particle implementation of the branching random walk. In contrast to existing signed-particle implementations, this weighted-particle one shows a key capacity of variance reduction by increasing the constant auxiliary function and has no time discretization errors. Several canonical numerical experiments on the 2D Gaussian barrier scattering and a 4D Helium-like system validate our theoretical findings, and demonstrate the accuracy, the efficiency, and thus the computability of the proposed weighted-particle Wigner branching random walk algorithm.
Abstract By designing a plasmonic waveguide–slit structure (a nanoslit etched in a silver nanowire) on a silver substrate, an ultrahigh Purcell factor and ultralarge figure of merit (FOM) are numerically predicted. Because of the large field enhancement (>150 times the incident field) and the ultrasmall optical volume (V ≈ 2 × 10−5λ3) of the resonant mode in the metallic nanoslit, the simulations show that the Purcell factor in the system can reach up to FP = 1.68 × 105, which is more than ten times the maximum Purcell factor in previous work (by placing metallic nanoparticles on a metal surface with a nanogap). Because of the utilization of a silver substrate rather than the common dielectric substrate, the mode cutoff of the surface plasmon polariton (SPP) waveguide mode is completely eliminated, which provides a large selection range of the nanowire radii to support the resonant mode in the nanoslit. Moreover, the SPP propagation length is significantly increased by more than 30 times. As a result, an ultralarge FOM of 1.40 × 107 is obtained, which is more than 80 times the maximum FOM in previous work where the metallic nanowire is placed on or surrounded by dielectric materials.
Carbon nanotubes (CNTs) and trace contaminants often co-occur in natural waters and wastewaters, and they may become the precursors of disinfection byproducts (DBPs). However, the effects of CNTs on the formation of DBPs during chlorination of co-existed organic pollutants are unknown. This study compared the effects of three types of CNTs on the formation of DBPs during chlorination of bisphenol A (BPA). The results showed that, compared with the single system of BPA, CNTs significantly decreased the initial rate (Ri) and the second-order rate constant (k) of trihalomethanes (THMs) formation in the binary systems of CNTs and BPA. For example, Ri for the binary system (38.7–49.6 µg/(L·h)) was much lower than that for the single system of BPA (63.1 µg/(L·h)). Furthermore, the suppression effects depended not only on the type but also on the concentration of CNTs: the suppression of Ri and k by CNTs followed the order of pristine CNTs > hydroxyl CNTs > carboxylic CNTs, and increased with rising concentration of CNTs. The adsorption experiments and density functional theory (DFT) calculation further revealed that higher adsorption and stronger binding of BPA to CNTs resulted in greater suppression degree of Ri and k by CNTs.
