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Surface lattice resonances (SLRs) in metasurfaces are promising in applications of sub-wavelength devices.Tunable and multimodal SLRs further enhance their appeal for flexible and multi-wavelength light-matter interactions. While multimodal SLRs offer promising properties, their realization often requires sophisticated designs, leading to limited tunability. Furthermore, current high-Q SLR implementations necessitate a homogeneous index in the operational environment, restricting potential applications such as biosensors that are typically operated in an aqueous or air cladding on a substrate. Here we present guided-SLRs (gSLRs) that are easily accessible in index-discontinuous environments, offering multimodal properties and straightforward tunability of resonances wavelengths, mode number, and mode coupling strengths. The gSLRs are achieved by coupling scattered light from metasurface units into a slab waveguide, creating a light ropagating channel in the lattice plane within an index-asymmetric environment. Tailoring the radiation pattern of individual units with guided transverse electric (TE) and transverse magnetic (TM) modes, multimodal resonances in both orthogonal and parallel coupling directions are accomplished. Mode number and mode frequency positions can be easily controlled by adjusting the waveguide configuration, while mode strength is tuned by vertical positions of lattices in the slab. Multimodal gSLRs with strong intensities and tunable ositions extending from visible to near-infrared range are achieved when compose metasurfaces with gold nanoparticle-on-mirror (NPoM) cavities. This easy-to-access, actively tunable and multimodal gSLR in inhomogeneous mediums will advance the realization of ultrathin and ultracompact nano-optical and optoelectronic devices.

Despite the success of electric vehicles on land, electrification of maritime ships is challenged by the dilemma of range anxiety and cargo-carrying capacity. The longer range requires larger batteries, which inevitably eat up the precious cargo space and weight. This paper breaks new ground by proposing a coordinated planning model for offshore charging stations (OCSs) and electric ships (ESs), marking a first in this field. Strategically situated OCS can partition a long maritime route into several shorter segments, which in turn lead to smaller batteries and thus larger cargo capacities. The research analyzed the impact of maritime geographical conditions on the placement and sizing process and provided insights into the trade-offs between battery size, cargo-carrying capacity, and the cruising range of different types of electrified ships. Using real Automatic Identification System (AIS) data, we estimated the economic feasibility of the Shanghai-Busan high-traffic maritime route and conducted a sensitivity analysis on factors affecting its economic viability. The results show that installing OCS can significantly reduce the propulsion cost compared with ESs without OCS and traditional internal combustion engine (ICE) ships.

Various powerful deep neural network architectures have made great contribution to the exciting successes of deep learning in the past two decades. Among them, deep Residual Networks (ResNets) are of particular importance because they demonstrated great usefulness in computer vision by winning the first place in many deep learning competitions. Also, ResNets were the first class of neural networks in the development history of deep learning that are really deep. It is of mathematical interest and practical meaning to understand the convergence of deep ResNets. We aim at characterizing the convergence of deep ResNets as the depth tends to infinity in terms of the parameters of the networks. Toward this purpose, we first give a matrix-vector description of general deep neural networks with shortcut connections and formulate an explicit expression for the networks by using the notions of activation domains and activation matrices. The convergence is then reduced to the convergence of two series involving infinite products of non-square matrices. By studying the two series, we establish a sufficient condition for pointwise convergence of ResNets. Our result is able to give justification for the design of ResNets. We also conduct experiments on benchmark machine learning data to verify our results.

Deep neural networks, as a powerful system to represent high dimensional complex functions, play a key role in deep learning. Convergence of deep neural networks is a fundamental issue in building the mathematical foundation for deep learning. We investigated the convergence of deep ReLU networks and deep convolutional neural networks in two recent researches (arXiv:2107.12530, 2109.13542). Only the Rectified Linear Unit (ReLU) activation was studied therein, and the important pooling strategy was not considered. In this current work, we study the convergence of deep neural networks as the depth tends to infinity for two other important activation functions: the leaky ReLU and the sigmoid function. Pooling will also be studied. As a result, we prove that the sufficient condition established in arXiv:2107.12530, 2109.13542 is still sufficient for the leaky ReLU networks. For contractive activation functions such as the sigmoid function, we establish a weaker sufficient condition for uniform convergence of deep neural networks., Comment: incomplete

With the increasing impact of low inertia due to the high penetration of distributed generation, virtual synchronous generator (VSG) technology has been proposed to improve the stability of the inverter-interfaced distributed generator by providing "virtual inertia". This paper presents a recent review of virtual inertia control covering significance, features, design principles, and state-of-art inertia strategies from both physical and mathematical perspectives to facilitate the wide application of the VSG. The definition and source of virtual inertia are given to help researchers to establish the concept of "virtual inertia". Then, this paper covers influencing mechanism studies of virtual inertia to reveal its functions. Also, a design framework of the virtual inertia is established by considering both the characteristics of the control system and the limitation of energy storage systems and renewable energy resources. Finally, several novel adaptive inertia control strategies are reviewed, and some aspects of potential future research are recommended.