A system of dual quaternion matrix equations with its applications

We employ the M-P inverses and ranks of quaternion matrices to establish the necessary and sufficient conditions for solving a system of the dual quaternion matrix equations $(AX, XC) = (B, D)$, along with providing an expression for its general solution. Serving as an application, we investigate the solutions to the dual quaternion matrix equations $AX = B$ and $XC=D$, including $\eta$-Hermitian solutions. Lastly, we design a numerical example to validate the main research findings of this paper

Individual position diversity in dependence socioeconomic networks increases economic output

The availability of big data recorded from massively multiplayer online role-playing games (MMORPGs) allows us to gain a deeper understanding of the potential connection between individuals' network positions and their economic outputs. We use a statistical filtering method to construct dependence networks from weighted friendship networks of individuals. We investigate the 30 distinct motif positions in the 13 directed triadic motifs which represent microscopic dependences among individuals. Based on the structural similarity of motif positions, we further classify individuals into different groups. The node position diversity of individuals is found to be positively correlated with their economic outputs. We also find that the economic outputs of leaf nodes are significantly lower than that of the other nodes in the same motif. Our findings shed light on understanding the influence of network structure on economic activities and outputs in socioeconomic system.Comment: 19 pages, 5 figure

Segatron: Segment-Aware Transformer for Language Modeling and Understanding

Transformers are powerful for sequence modeling. Nearly all state-of-the-art language models and pre-trained language models are based on the Transformer architecture. However, it distinguishes sequential tokens only with the token position index. We hypothesize that better contextual representations can be generated from the Transformer with richer positional information. To verify this, we propose a segment-aware Transformer (Segatron), by replacing the original token position encoding with a combined position encoding of paragraph, sentence, and token. We first introduce the segment-aware mechanism to Transformer-XL, which is a popular Transformer-based language model with memory extension and relative position encoding. We find that our method can further improve the Transformer-XL base model and large model, achieving 17.1 perplexity on the WikiText-103 dataset. We further investigate the pre-training masked language modeling task with Segatron. Experimental results show that BERT pre-trained with Segatron (SegaBERT) can outperform BERT with vanilla Transformer on various NLP tasks, and outperforms RoBERTa on zero-shot sentence representation learning.Comment: Accepted by AAAI 202

Quantifying immediate price impact of trades based on the kk-shell decomposition of stock trading networks

Traders in a stock market exchange stock shares and form a stock trading network. Trades at different positions of the stock trading network may contain different information. We construct stock trading networks based on the limit order book data and classify traders into $k$ classes using the $k$-shell decomposition method. We investigate the influences of trading behaviors on the price impact by comparing a closed national market (A-shares) with an international market (B-shares), individuals and institutions, partially filled and filled trades, buyer-initiated and seller-initiated trades, and trades at different positions of a trading network. Institutional traders professionally use some trading strategies to reduce the price impact and individuals at the same positions in the trading network have a higher price impact than institutions. We also find that trades in the core have higher price impacts than those in the peripheral shell.Comment: 6 pages including 3 figures and 1 tabl