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A term called the quasi-projection pair $(P,Q)$ was introduced recently by the authors, where $P$ is a projection and $Q$ is an idempotent on a Hilbert $C^*$-module $H$ satisfying $Q^*=(2P-I)Q(2P-I)$, in which $Q^*$ is the adjoint operator of the idempotent $Q$ and $I$ is the identity operator on $H$. Some fundamental issues on quasi-projection pairs, such as the block matrix representations for quasi-projection pairs and the $C^*$-morphisms associated with quasi-projection pairs, are worthwhile to be investigated. This paper aims to make some preparations. One object called the semi-harmonious quasi-projection pair is introduced in the general setting of the adjointable operators on Hilbert $C^*$-modules. Some related operator theories on the common similarity of operators and a norm equation associated with the Friedrichs angle are dealt with., Comment: arXiv admin note: substantial text overlap with arXiv:2305.12984

The main purpose of this paper is to give a full characterization of the operator distances from projections to an idempotent, which includes the minimum value, the maximum value and the intermediate values. Let $H$ be a Hilbert space and $\mathbb{B}(H)$ be the set of bounded linear operators on $H$. Given an arbitrary idempotent $Q\in \mathbb{B}(H)$, it is proved that $$\|m(Q)-Q\|\le \|P-Q\|\le \|I-m(Q)-Q\|$$ for every projection $P$ on $H$, in which $I$ is the identity operator on $H$ and $m(Q)$ is a specific projection called the matched projection of $Q$. When $Q\in\mathbb{B}(H)$ is a non-projection idempotent, it is proved that for every number $\alpha$ contained in the interval $\left[\|m(Q)-Q\|,\|I-m(Q)-Q\|\right]$, there exists a projection $P\in \mathbb{B}(H)$ such that $\|P-Q\|=\alpha$. Two uniqueness problems concerning the projections that attain the minimum value or the maximum value are also dealt with.

Task-oriented dialogue systems are expected to handle a constantly expanding set of intents and domains even after they have been deployed to support more and more functionalities. To live up to this expectation, it becomes critical to mitigate the catastrophic forgetting problem (CF) that occurs in continual learning (CL) settings for a task such as intent recognition. While existing dialogue systems research has explored replay-based and regularization-based methods to this end, the effect of domain ordering on the CL performance of intent recognition models remains unexplored. If understood well, domain ordering has the potential to be an orthogonal technique that can be leveraged alongside existing techniques such as experience replay. Our work fills this gap by comparing the impact of three domain-ordering strategies (min-sum path, max-sum path, random) on the CL performance of a generative intent recognition model. Our findings reveal that the min-sum path strategy outperforms the others in reducing catastrophic forgetting when training on the 220M T5-Base model. However, this advantage diminishes with the larger 770M T5-Large model. These results underscores the potential of domain ordering as a complementary strategy for mitigating catastrophic forgetting in continually learning intent recognition models, particularly in resource-constrained scenarios.

This paper deals mainly with some aspects of the adjointable operators on Hilbert $C^*$-modules. A new tool called the generalized polar decomposition for each adjointable operator is introduced and clarified. As an application, the general theory of the weakly complementable operators is set up in the framework of Hilbert $C^*$-modules. It is proved that there exists an operator equation which has a unique solution, whereas this unique solution fails to be the reduced solution. Some investigations are also carried out in the Hilbert space case. It is proved that there exist a closed subspace $M$ of certain Hilbert space $K$ and an operator $T\in \mathbb{B}(K)$ such that $T$ is $(M,M)$-weakly complementable, whereas $T$ fails to be $(M,M)$-complementable. The solvability of the equation $$A:B=X^*AX+(I-X)^*B(I-X) \quad (X\in\mathbb{B}(H))$$ is also dealt with in the Hilbert space case, where $A,B\in \mathbb{B}(H)$ are two general positive operators, and $A:B$ denotes their parallel sum. Among other things, it is shown that there exist certain positive operators $A$ and $B$ on the Hilbert space $\ell^2(\mathbb{N})\oplus \ell^2(\mathbb{N})$ such that the above equation has no solution., Comment: This version is accepted for publication in Banach Journal of Mathematical Analysis

For each pair of matrices $A$ and $B$ with the same order, let $\|A-B\|_F$ denote their Frobenius distance. This paper deals mainly with the Frobenius distances from projections to an idempotent matrix. For every idempotent $Q\in \mathbb{C}^{n\times n}$, a projection $m(Q)$ called the matched projection can be induced. It is proved that $m(Q)$ is the unique projection whose Frobenius distance away from $Q$ takes the minimum value among all the Frobenius distances from projections to $Q$, while $I_n-m(Q)$ is the unique projection whose Frobenius distance away from $Q$ takes the maximum value. Furthermore, it is proved that for every number $\alpha$ between the minimum value and the maximum value, there exists a projection $P$ whose Frobenius distance away from $Q$ takes the value $\alpha$. Based on the above characterization of the minimum distance, some Frobenius norm upper bounds and lower bounds of $\|P-Q\|_F$ are derived under the condition of $PQ=Q$ on a projection $P$ and an idempotent $Q$.

Natural Language Processing (NLP) plays a significant role in our daily lives and has become an essential part of Artificial Intelligence (AI) education in K-12. As children grow up with NLP-powered applications, it is crucial to introduce NLP concepts to them, fostering their understanding of language processing, language generation, and ethical implications of AI and NLP. This paper presents a comprehensive review of digital learning environments for teaching NLP in K-12. Specifically, it explores existing digital learning tools, discusses how they support specific NLP tasks and procedures, and investigates their explainability and evaluation results in educational contexts. By examining the strengths and limitations of these tools, this literature review sheds light on the current state of NLP learning tools in K-12 education. It aims to guide future research efforts to refine existing tools, develop new ones, and explore more effective and inclusive strategies for integrating NLP into K-12 educational contexts., Comment: 24 pages, 13 figures

The aim of this paper is to give new characterizations of some fundamental issues about idempotents. In the general setting of adjointable operators on Hilbert $C^*$-modules, a new term of quasi-projection pair is introduced. For each idempotent $Q$, a projection $m(Q)$, called the matched projection of $Q$, is constructed. It is shown that $Q$ and $m(Q)$ as idempotents are homotopic, and $\big(m(Q),Q\big)$ is a quasi-projection pair. Some formulas for $m(Q)$ are derived. Based on these formulas, representations and norm estimations associated with $m(Q)$ are dealt with., Comment: The original submission entitled "Quasi-projection pairs on Hilbert $C^*$-modules" will be divided into three parts, and this is the first part. Some minor changes have been made in this revision, which is accepted for publication in Journal of Operator Theory

Power dynamics in human-human communication can impact rapport-building and learning gains, but little is known about how power impacts human-agent communication. In this paper, we examine dominance behavior in utterances between middle-school students and a teachable robot as they work through math problems, as coded by Rogers and Farace's Relational Communication Control Coding Scheme (RCCCS). We hypothesize that relatively dominant students will show increased learning gains, as will students with greater dominance agreement with the robot. We also hypothesize that gender could be an indicator of difference in dominance behavior. We present a preliminary analysis of dominance characteristics in some of the transactions between robot and student. Ultimately, we hope to determine if manipulating the dominance behavior of a learning robot could support learning., Comment: 8 pages, 1 figure