For a positive integer $N$, let $mathscr{C}_N(mathbb{Q})$ be the rational cuspidal subgroup of $J_0(N)$ and $mathscr{C}(N)$ be the rational cuspidal divisor class group of $X_0(N)$, which are both subgroups of the rational torsion subgroup of $J_0(N)
$. We prove that two groups $mathscr{C}_N(mathbb{Q})$ and $mathscr{C}(N)$ are equal when $N=p^2M$ for any prime $p$ and any squarefree integer $M$. To achieve this we show that all modular units on $X_0(N)$ can be written as products of certain functions $F_{m, h}$, which are constructed from generalized Dedekind eta functions. Also, we determine the necessary and sufficient conditions for such products to be modular units on $X_0(N)$ under a mild assumption.
Graph neural networks (GNNs), which learn the node representations by recursively aggregating information from its neighbors, have become a predominant computational tool in many domains. To handle large-scale graphs, most of the existing methods par
tition the input graph into multiple sub-graphs (e.g., through node clustering) and apply batch training to save memory cost. However, such batch training will lead to label bias within each batch, and then result in over-confidence in model predictions. Since the connected nodes with positively related labels tend to be assigned together, the traditional cross-entropy minimization process will attend on the predictions of biased classes in the batch, and may intensify the overfitting issue. To overcome the label bias problem, we propose the adaptive label smoothing (ALS) method to replace the one-hot hard labels with smoothed ones, which learns to allocate label confidences from the biased classes to the others. Specifically, ALS propagates node labels to aggregate the neighborhood label distribution in a pre-processing step, and then updates the optimal smoothed labels online to adapt to specific graph structure. Experiments on the real-world datasets demonstrate that ALS can be generally applied to the main scalable learning frameworks to calibrate the biased labels and improve generalization performances.
Wide field-of-view (FOV) optics are widely used in various imaging, display, and sensing applications. While conventional wide FOV optics rely on cascading multiple elements to suppress coma and other aberrations, it has recently been demonstrated th
at diffraction-limited, near-180 degree FOV operation can be achieved with a single-piece flat fisheye lens designed via iterative numerical optimization [Nano Lett. 20, 7429(2020)]. Here we derive an analytical solution to enable computationally efficient design of flat wide FOV lenses based on metasurfaces or diffractive optical elements (DOEs). Leveraging this analytical approach, we further quantified trade-offs between optical performance and design parameters in wide FOV metalenses.
Motion forecasting plays a significant role in various domains (e.g., autonomous driving, human-robot interaction), which aims to predict future motion sequences given a set of historical observations. However, the observed elements may be of differe
nt levels of importance. Some information may be irrelevant or even distracting to the forecasting in certain situations. To address this issue, we propose a generic motion forecasting framework (named RAIN) with dynamic key information selection and ranking based on a hybrid attention mechanism. The general framework is instantiated to handle multi-agent trajectory prediction and human motion forecasting tasks, respectively. In the former task, the model learns to recognize the relations between agents with a graph representation and to determine their relative significance. In the latter task, the model learns to capture the temporal proximity and dependency in long-term human motions. We also propose an effective double-stage training pipeline with an alternating training strategy to optimize the parameters in different modules of the framework. We validate the framework on both synthetic simulations and motion forecasting benchmarks in different domains, demonstrating that our method not only achieves state-of-the-art forecasting performance, but also provides interpretable and reasonable hybrid attention weights.
Opinion prediction is an emerging research area with diverse real-world applications, such as market research and situational awareness. We identify two lines of approaches to the problem of opinion prediction. One uses topic-based sentiment analysis
with time-series modeling, while the other uses static embedding of text. The latter approaches seek user-specific solutions by generating user fingerprints. Such approaches are useful in predicting users reactions to unseen content. In this work, we propose a novel dynamic fingerprinting method that leverages contextual embedding of users comments conditioned on relevant users reading history. We integrate BERT variants with a recurrent neural network to generate predictions. The results show up to 13% improvement in micro F1-score compared to previous approaches. Experimental results show novel insights that were previously unknown such as better predictions for an increase in dynamic history length, the impact of the nature of the article on performance, thereby laying the foundation for further research.
We raise a detuning-dependent loss mechanism to describe the soliton formation dynamics when the lumped filtering operation is manipulated in anomalous group velocity dispersion regime, using stability analysis of generalized Lugiato-Lefever equation.
On-chip manipulation of single resonance over broad background comb spectra of microring resonators is indispensable, ranging from tailoring laser emission, optical signal processing to non-classical light generation, yet challenging without scarifyi
ng the quality factor or inducing additional dispersive effects. Here, we propose an experimentally feasible platform to realize on-chip selective depletion of single resonance in microring with decoupled dispersion and dissipation, which are usually entangled by Kramer-Kroning relation. Thanks to the existence of non-Hermitian singularity, unsplit but significantly increased dissipation of the selected resonance is achieved due to the simultaneous collapse of eigenvalues and eigenvectors, fitting elegantly the requirement of pure single-mode depletion. With delicate yet experimentally feasible parameters, we show explicit evidence of modulation instability as well as deterministic single soliton generation in microresonators induced by depletion in normal and anomalous dispersion regime, respectively. Our findings connect non-Hermitian singularities to wide range of applications associated with selective single mode manipulation in microwave photonics, quantum optics, ultrafast optics and beyond.
In causal mediation studies that decompose an average treatment effect into a natural indirect effect (NIE) and a natural direct effect (NDE), examples of post-treatment confounding are abundant. Past research has generally considered it infeasible t
o adjust for a post-treatment confounder of the mediator-outcome relationship due to incomplete information: it is observed under the actual treatment condition while missing under the counterfactual treatment condition. This study proposes a new sensitivity analysis strategy for handling post-treatment confounding and incorporates it into weighting-based causal mediation analysis without making extra identification assumptions. Under the sequential ignorability of the treatment assignment and of the mediator, we obtain the conditional distribution of the post-treatment confounder under the counterfactual treatment as a function of not just pretreatment covariates but also its counterpart under the actual treatment. The sensitivity analysis then generates a bound for the NIE and that for the NDE over a plausible range of the conditional correlation between the post-treatment confounder under the actual and that under the counterfactual conditions. Implemented through either imputation or integration, the strategy is suitable for binary as well as continuous measures of post-treatment confounders. Simulation results demonstrate major strengths and potential limitations of this new solution. A re-analysis of the National Evaluation of Welfare-to-Work Strategies (NEWWS) Riverside data reveals that the initial analytic results are sensitive to omitted post-treatment confounding.
We investigate the sideband spectra of a driven nonlinear mode with its eigenfrequency being modulated at a low frequency (< 1 kHz). This additional parametric modulation leads to prominent antiresonance lineshapes in the sideband spectra, which can
be controlled through the vibration state of the driven mode. We also establish a direct connection between the antiresonance frequency and the squeezing of thermal fluctuation in the system. Our work not only provides a simple and robust method for squeezing characterization but also opens a new possibility toward sideband applications.
We consider Greens functions $G(z):=(H-z)^{-1}$ of Hermitian random band matrices $H$ on the $d$-dimensional lattice $(mathbb Z/Lmathbb Z)^d$. The entries $h_{xy}=overline h_{yx}$ of $H$ are independent centered complex Gaussian random variables with
variances $s_{xy}=mathbb E|h_{xy}|^2$. The variances satisfy a banded profile so that $s_{xy}$ is negligible if $|x-y|$ exceeds the band width $W$. For any $nin mathbb N$, we construct an expansion of the $T$-variable, $T_{xy}=|m|^2 sum_{alpha}s_{xalpha}|G_{alpha y}|^2$, with an error $O(W^{-nd/2})$, and use it to prove a local law on the Greens function. This $T$-expansion was the main tool to prove the delocalization and quantum diffusion of random band matrices for dimensions $dge 8$ in part I of this series.
