Representation Stability and Finite Orthogonal Groups

In this paper, we prove stability results about orthogonal groups over finite commutative rings where 2 is a unit. Inspired by Putman and Sam (2017), we construct a category $\mathbf{OrI}(R)$ and prove a Noetherianity theorem for the category of $\mathbf{OrI}(R)$-modules. This implies an asymptotic structure theorem for orthogonal groups. In addition, we show general homological stability theorems for orthogonal groups, with both untwisted and twisted coefficients, partially generalizing a result of Charney (1987).Comment: 21 pages, 0 figure

Leader-following consensus for lower-triangular nonlinear multi-agent systems with unknown controller and measurement sensitivities

summary:In this paper, a novel consensus algorithm is presented to handle with the leader-following consensus problem for lower-triangular nonlinear MASs (multi-agent systems) with unknown controller and measurement sensitivities under a given undirected topology. As distinguished from the existing results, the proposed consensus algorithm can tolerate to a relative wide range of controller and measurement sensitivities. We present some important matrix inequalities, especially a class of matrix inequalities with multiplicative noises. Based on these results and a dual-domination gain method, the output consensus error with unknown measurement noises can be used to construct the compensator for each follower directly. Then, a new distributed output feedback control is designed to enable the MASs to reach consensus in the presence of large controller perturbations. In view of a Lyapunov function, sufficient conditions are presented to guarantee that the states of the leader and followers can achieve consensus asymptotically. In the end, the proposed consensus algorithm is tested and verified by an illustrative example

Complexity of emerging magnetic flux during lifetime of solar ephemeral regions

As a relatively active region, ephemeral region (ER) exhibits highly complex pattern of magnetic flux emergence. We aim to study detailed secondary flux emergences (SFEs) which we define as bipoles that they appear close to ERs and finally coalesce with ERs after a period. We study the SFEs during the whole process from emergence to decay of 5 ERs observed by the Helioseismic and Magnetic Imager (HMI) aboard Solar Dynamics Observatory (SDO) . The maximum unsigned magnetic flux for each ER is around $10^{20}$ Mx. Each ER has tens of SFEs with an average emerging magnetic flux of approximately 5$\times10^{18}$ Mx. The frequency of normalized magnetic flux for all the SFEs follows a power law distribution with an index of -2.08 . The majority of SFEs occur between the positive and negative polarities of ER , and their growth time is concentrated within one hour. The magnetic axis of SFE is found to exhibit a random distribution in the 5 ERs. We suggest that the relationship between SFEs and ERs can be understood by regarding the photospheric magnetic field observations as cross-sections of an emerging magnetic structure. Tracking the ERs' evolution, we propose that these SFEs in ERs may be sequent emergences from the bundle of flux tube of ERs, and that SFEs are partially emerged $\Omega$-loops.Comment: 12 pages, 9 figures, 1 table and accepted for publication in the Astrophysical Journa

Automatic Answerability Evaluation for Question Generation

Conventional automatic evaluation metrics, such as BLEU and ROUGE, developed for natural language generation (NLG) tasks, are based on measuring the n-gram overlap between the generated and reference text. These simple metrics may be insufficient for more complex tasks, such as question generation (QG), which requires generating questions that are answerable by the reference answers. Developing a more sophisticated automatic evaluation metric, thus, remains as an urgent problem in QG research. This work proposes a Prompting-based Metric on ANswerability (PMAN), a novel automatic evaluation metric to assess whether the generated questions are answerable by the reference answers for the QG tasks. Extensive experiments demonstrate that its evaluation results are reliable and align with human evaluations. We further apply our metric to evaluate the performance of QG models, which shows our metric complements conventional metrics. Our implementation of a ChatGPT-based QG model achieves state-of-the-art (SOTA) performance in generating answerable questions

CrossVideo: Self-supervised Cross-modal Contrastive Learning for Point Cloud Video Understanding

This paper introduces a novel approach named CrossVideo, which aims to enhance self-supervised cross-modal contrastive learning in the field of point cloud video understanding. Traditional supervised learning methods encounter limitations due to data scarcity and challenges in label acquisition. To address these issues, we propose a self-supervised learning method that leverages the cross-modal relationship between point cloud videos and image videos to acquire meaningful feature representations. Intra-modal and cross-modal contrastive learning techniques are employed to facilitate effective comprehension of point cloud video. We also propose a multi-level contrastive approach for both modalities. Through extensive experiments, we demonstrate that our method significantly surpasses previous state-of-the-art approaches, and we conduct comprehensive ablation studies to validate the effectiveness of our proposed designs

Local Convergence of Approximate Newton Method for Two Layer Nonlinear Regression

There have been significant advancements made by large language models (LLMs) in various aspects of our daily lives. LLMs serve as a transformative force in natural language processing, finding applications in text generation, translation, sentiment analysis, and question-answering. The accomplishments of LLMs have led to a substantial increase in research efforts in this domain. One specific two-layer regression problem has been well-studied in prior works, where the first layer is activated by a ReLU unit, and the second layer is activated by a softmax unit. While previous works provide a solid analysis of building a two-layer regression, there is still a gap in the analysis of constructing regression problems with more than two layers. In this paper, we take a crucial step toward addressing this problem: we provide an analysis of a two-layer regression problem. In contrast to previous works, our first layer is activated by a softmax unit. This sets the stage for future analyses of creating more activation functions based on the softmax function. Rearranging the softmax function leads to significantly different analyses. Our main results involve analyzing the convergence properties of an approximate Newton method used to minimize the regularized training loss. We prove that the loss function for the Hessian matrix is positive definite and Lipschitz continuous under certain assumptions. This enables us to establish local convergence guarantees for the proposed training algorithm. Specifically, with an appropriate initialization and after $O(\log(1/\epsilon))$ iterations, our algorithm can find an $\epsilon$-approximate minimizer of the training loss with high probability. Each iteration requires approximately $O(\mathrm{nnz}(C) + d^\omega)$ time, where $d$ is the model size, $C$ is the input matrix, and $\omega < 2.374$ is the matrix multiplication exponent