An \( (n,r) \)-arc in \( \operatorname{PG}(2,q) \) is a set \( \mathcal{B} \) of points in \( \operatorname{PG}(2,q) \) such that each line in \( \operatorname{PG}(2,q) \) contains at most \( r \) elements of \( \mathcal{B} \) and such that there is at least one line containing exactly \( r \) elements of \( \mathcal{B} \). The value \( m_r(2,q) \) denotes the maximal number \( n \) of points in the projective geometry \( \operatorname{PG}(2,q) \) for which an \( (n,r) \)-arc exists. We show by systematically excluding possible automorphisms that putative \( (44,5) \)-arcs, \( (90,9) \)-arcs in \( \operatorname{PG}(2,11) \), and \( (39,4) \)-arcs in \( \operatorname{PG}(2,13) \)—in case of their existence—are rigid, i.e. they all would only admit the trivial automorphism group of order \( 1 \). In addition, putative \( (50,5) \)-arcs, \( (65,6) \)-arcs, \( (119,10) \)-arcs, \( (133,11) \)-arcs, and \( (146,12) \)-arcs in \( \operatorname{PG}(2,13) \) would be rigid or would admit a unique automorphism group (up to conjugation) of order \( 2 \).

Let \( S \) be a connected union of finitely many \( d \)-dimensional boxes in \( \mathbb{R}^d \) and let \( \mathcal{B} \) represent the family of boxes determined by facet hyperplanes for \( S \), with \( \mathcal{F} \) the associated family of faces (including members of \( \mathcal{B} \)). For set \( F \) in \( \mathcal{F} \), point \( x \) relatively interior to \( F \), and point \( y \) in \( S \), \( x \) sees \( y \) via staircase paths in \( S \) if and only if every point of \( F \) sees \( y \) via such paths. Thus the visibility set of \( x \) is a union of members of \( \mathcal{F} \), as is the staircase kernel of \( S \). A similar result holds for \( k \)-staircase paths in \( S \) and the \( k \)-staircase kernel of \( S \).

The minimum dominating set problem asks for a dominating set with minimum size. First, we determine some vertices contained in the minimum dominating set of a graph. By applying a particular scheme, we ensure that the resulting graph is 2-connected and the length of each formed induced cycle is 0 mod 3. We label every three vertices in the induced cycles of length 0 mod 3. Then there is a way of labeling in which the set of all labeled vertices is the minimum dominating set of the resulting graph, and is contained in the minimum dominating set of the original graph. We also consider the remaining vertices of the minimum dominating set of the original graph and determine all vertices contained in the minimum dominating set of a graph with maximum degree 3. The complexity of the minimum dominating set problem for cubic graphs was shown to be APX-complete in 2000 and this problem is solved by our arguments in polynomial time.

In this paper we study a new graph parameter, the stacking number. Defined in relation to the eternal domination game, we show that there are highly connected graphs for which it is beneficial to allow multiple guards to occupy a vertex, answering an open question of Finbow et al. In fact, we show that for any sequence \( (s_i) \), allowing \( s_j \) guards to occupy a vertex can save arbitrarily many guards in comparison to allowing fewer than this on a vertex. We also show that the stacking number is \( 1 \) for all trees.

The body language of dancers is vital for conveying emotion. In this study, Kinect is used to detect and track dancers’ movements, and we develop two models: a dance action recognition model based on skeleton data and a dance emotion recognition model using an Attention-ConvLSTM. The action recognition model achieves 88.34% accuracy—reaching its best performance after just 40 iterations—while the emotion recognition model reaches an accuracy of 98.95%. Our analysis shows that features such as eigenvalue speed, skeleton pair distance, and inclination effectively differentiate emotions, although certain emotions (e.g., Excited vs. Pleased and Relaxed vs. Sad) can be confused. Notably, the leg’s skeletal points significantly influence emotion expression. Ultimately, the study establishes a dance emotion expression mechanism through coordinated movement changes of the head, hands, legs, waist, and torso.

Deep learning-based target detection algorithms outperform traditional methods by eliminating the need for manual feature design and improving accuracy and efficiency. This paper constructs a YOLOv5 target detection model using a deep convolutional neural network. To enhance accuracy, generalization, and detection speed, three data augmentation techniques—mosaic data enhancement, adaptive anchor frame, and adaptive image scaling—are applied. The model is further optimized with an attention mechanism and a modified YOLOv5 framework. A loss function and global average pooling enhance feature mapping for a fully convolutional network. Experimental results show that the improved YOLOv5n model achieves a 2.9979 percentage point increase in MAP, a 31% improvement in FPS, and a training time reduction of 10 minutes, completing 100 rounds in 20 minutes.

Metacognition, as a fundamental ability for learners to adapt to complex environments, is equally adapted to constructivist teaching and learning activities. In this paper, we propose a model of learning environment characteristics for metacognitive regulation under constructivist learning theory, and utilize Item2Vec algorithm, Self-Attention mechanism, and BiGRU model to construct a model of metacognitive ability. The model presents a kind of multi-channel network characteristic composed of Self-Attention mechanism and BiGRU model. Design a theoretical model of the learning environment oriented to improving students’ metacognitive ability, and analyze the functional modules of the overall system of the learning environment. Propose a learning activity aiming at the improvement of metacognitive ability and incorporating constructivist theory as the guiding concept to allocate the various aspects of the whole constructivist teaching activity. Analyze the implementation effect of constructivist teaching activities based on metacognitive strategies and organize the influencing factors of metacognitive strategies. The bivariate correlation analysis of students’ total test scores and usual grades are closely related to planning strategies, monitoring strategies, and regulating strategies, and the significance (two-tailed) is less than 0.01. This indicates that the higher the students’ scores, the higher the corresponding level of metacognitive strategies.

Civil engineering crack detection faces challenges due to complex environments and external interferences. This paper proposes an improved YOLO v8s-WOMA network, integrating ODConv, C2f-MA modules, and WIoU loss function to enhance crack identification accuracy. A BP neural network is also trained to assess crack damage. Experiments on the CBP dataset compare this method with existing detection algorithms. Results show that the proposed model achieves the highest mAP (90.5%), F1-score (90.3%), and accuracy (89.6%). Bridge crack detection errors remain within 0.1mm (width) and 20mm (length), ensuring precise damage assessment. The model effectively handles complex backgrounds, accurately detects cracks, and meets practical engineering needs.

The rapid growth of multilingual information online has made traditional translation insufficient, highlighting the need for intelligent language translation. This study employs a convolutional neural network to extract visual features from translated images and uses region-selective attention to align text and image features. The fused information is then processed through a sequence model to develop a computer vision-based translation algorithm. Results show that the proposed algorithm excels in key evaluation metrics, improving translation quality. It maintains a low leakage rate (1.30%), a mistranslation rate of 2.64%, and an average response time of 67.28ms. With strong generalization and applicability in multilingual translation, the algorithm demonstrates high performance and promising real-world applications.

This paper addresses the limitations of the traditional portfolio theory centered on the mean-variance model and expected utility theory, and proposes the establishment of a portfolio model that takes into account the subjective psychological factors of investors, taking into account the fact that investors are susceptible to the influence of various psychological biases, affective biases, and cognitive biases in the actual decision-making process, with respect to the theory of consistency of the assumptions of the investor’s risk attitude. The portfolio model based on fuzzy decision-making is proposed, combined with the development and application of linear programming in portfolio optimization, the return of assets is regarded as a random fuzzy variable, and the stochastic fuzzy portfolio model is constructed to consider the risk characteristics of investors. The portfolio returns under different emotions or different risk preferences are explored separately. Combined with the fund categorization allocation of the sample firms, the fund portfolio C based on the fuzzy portfolio model is proposed and compared with the equal weight allocation fund (fund portfolio A) and the risk coefficient weighted allocation fund (fund portfolio B) based on the risk level of return, respectively. Fund Portfolio C has the highest average return.