For every connected graph \(F\) with \(n\) vertices and every graph \(G\) with chromatic surplus \(s(G)\leq n\), the Ramsey number \(r(F,G)\) satisfies \(
r(F,G) \geq (n-1)(\chi(G)-1) + s(G), \) where \(\chi(G)\) denotes the chromatic number of \(G\). If this lower bound is attained, then \(F\) is called \(G\)-good. For all connected graphs \(G\) with at most six vertices and \(\chi(G) \geq 4\), every tree \(T_n\) of order \(n\geq 5\) is \(G\)-good. In case of \(\chi(G) = 3\) and \(G \neq K_6-3K_2\), every non-star tree \(T_n\) is \(G\)-good except for some small \(n\), whereas \(r(S_n,G)\) for the star \(S_n = K_{1,n-1}\) in a few cases differs by at most 2 from the lower bound. In this note, we prove that the values of \(r(S_n,K_6-3K_2)\) are considerably larger for sufficiently large \(n\). Furthermore, exact values of \(r(S_n,K_6-3K_2)\) are obtained for small \(n\).

Let \(A\) be a real algebra. It is called locally complex algebra if every non-zero element generates a subalgebra isomorphic to either \(\mathbb{R}\) or \(\mathbb{C}.\) It is said to satisfy the  uniqueness of the square root except the sign if the equation \(x^2=y^2\) implies \(y=\pm x.\) We show the following:
1. Every locally complex algebra is a quadratic algebra.
2. Every alternative locally complex algebra is isomorphic to either \(\mathbb{R},\) \(\mathbb{C},\) \(\mathbb{H}\) or \(\mathbb{O}.\)
3. Every commutative locally complex algebra without divisors of zero is isomorphic to \(\mathbb{R}\) or \(\mathbb{C}.\)
4. Every finite-dimensional algebra satisfying the uniqueness of the square root except the sign has dimension \(\leq 2\) and contains non-zero idempotents.

To solve the problem of identifying intrinsic relationships between objects and mirror segmentation in semantic segmentation of urban scenes using current multi-modal data, this study innovatively integrates color images, depth information, and thermal images to propose a network model that integrates modal memory sharing and form complementarity, and a hierarchical assisted fusion network model. Compared with existing advanced urban scene semantic segmentation methods, the proposed method performed excellently in terms of performance, with an average pixel accuracy and mean intersection over union of over 80% for different objects. In addition, the research method achieved clearer and more complete segmentation results by strengthening contextual associations, and edge processing is also smoother. Even in object segmentation with similarities in distance, shape, and brightness such as “vegetation” and “sidewalk”, the research method still maintained high accuracy. The research method can effectively handle the complexity of urban scenes, providing a new solution for semantic segmentation of multi-modal data in urban scenes.

Blockchain technology has the characteristics of data anti-tampering and anti-forgery, which can provide solution ideas for the secure storage and transmission of data in distributed networks. The study applies blockchain technology to data auditing, constructs an aggregated signature based on conditional identity anonymization to protect user privacy, simplifies the auditing computation by using homomorphic hash function, and deploys three kinds of smart contracts on the blockchain to design a blockchain-based data integrity auditing scheme. For the privacy protection problem, a blockchain privacy protection model based on differential privacy is constructed by integrating the differential privacy policy into the blockchain smart contract layer. The experimental results show that the data integrity auditing scheme has superior blockchain storage cost and time overhead, and the average time overhead under different dynamic operations is below 30ms. The privacy protection model also exhibits high efficiency, with encryption and decryption times of 0.075s and 0.063s, respectively, under the largest data file, and a significant speed advantage in all phases of operation. The proposed scheme in this paper meets the needs of data integrity and privacy protection, and can provide efficient services for users.

Mango weaving, a traditional handicraft in Guangxi, is facing decline. This study explores AI technology’s role in its protection and innovation by analyzing consumer reviews using perceived value theory and the LDA topic model to identify preferences for improving production. A lightweight generative adversarial network with a non-local attention mechanism is proposed for text-to-multi-objective image generation, aiding innovative design. Consumers expressed 82.6% satisfaction with mango weaving. Reviews were categorized into five themes, highlighting the need for improvements in emotion, quality, and price. The AI-generated image model outperformed others, with IS and FID scores improving by 21.85% and 16.46%, respectively. AI enhances mango weaving by refining design, improving product quality, and expanding its preservation and development.

The ocean is vital for human survival and development, serving as the birthplace of life and a source of food, minerals, and scientific research materials. It plays a crucial role in global trade, economic growth, climate regulation, and ecological balance. Underwater positioning technology is fundamental to marine engineering, with underwater acoustic passive positioning being essential for sonar source localization. Active and passive acoustic systems help measure underwater noise and determine target locations. Passive systems rely on signals emitted by targets, while active systems use interaction signals for positioning. This study applies machine learning to improve acoustic beacon signal recognition in underwater positioning. Results show that machine learning enhances recognition speed by 8% and detection accuracy by 9% compared to traditional methods. By optimizing underwater acoustic signal recognition, this approach enhances positioning accuracy, reduces costs, and advances intelligent marine technology, providing innovative solutions for complex marine environments.

Nowadays, “Artificial Intelligence + Education” is transforming teaching and learning. In this study, we employ AI-based data mining to innovate educational management by designing an academic monitoring system using K-means clustering and developing an early warning model through stacking multi-model superposition. Targeted management measures, including personalized video recommendations, are implemented based on the model’s predictions to promote individualized student development. By analyzing daily behavior data from 500 college students, the K-means algorithm effectively classified them into four groups, and the academic alert model achieved a prediction accuracy of 84.19%, outperforming single base models. The implementation of this personalized management method significantly improved student performance compared to traditional approaches, demonstrating its potential to enhance educational outcomes.

Let \( p \) be a prime number, and let \( k \) and \( m \) be positive integers with \( k \geq 2 \). This paper studies the algebraic structure of \(\lambda\)-constacyclic codes of arbitrary length over the finite commutative ring \( R = \frac{\mathbb{F}_{p^m}[u, v]}{ \langle u^k, v^2, uv – vu \rangle } \), where \(\lambda\) is a unit in \( R \) given by \( \lambda = \sum\limits_{i=0}^{k-1} \lambda_i u^i + v\sum\limits_{i=0}^{k-1} \lambda_i’ u^i \), with \(\lambda_i, \lambda_i’ \in \mathbb{F}_{p^m}\) and \(\lambda_0, \lambda_1 \neq 0\). We provide a complete classification of these constacyclic codes, determine their dual structures, and compute their Hamming distances when the code length is \( p^s \).

In this paper, the hyperoctahedral group algebra \(\mathscr{F}[\overrightarrow{S_{n}}]\) over a splitting field \(\mathscr{F}\) of wreath product \(\overrightarrow{S_{n}}\) with \(\text{char}(\mathscr{F})\nmid|\overrightarrow{S_{n}}|\), is considered and the unique idempotents corresponding to the four linear characters of the group \(\overrightarrow{S_{n}}\) are explored. Also, by establishing the minimum weights and dimensions, all group codes generated by the linear idempotents in the aforementioned group algebra are completely characterized for every \(n\). The nonlinear idempotents corresponding to nonlinear characters of \(\overrightarrow{S_{3}}\) are also obtained and various group codes in \(\mathscr{F}[\overrightarrow{S_{3}}]\) generated by linear and nonlinear idempotents are examined.

Let \( G = (V, E) \) be a graph with minimum degree at least one. The general inverse degree of \( G \) is defined as \(\sum\limits_{v \in V} \frac{1}{d^{\alpha}(v)}\), where \( \alpha \) is a real number with \( \alpha > 0 \). In this paper, we present sufficient conditions involving the general inverse degree with \( \alpha \geq 1 \) for some Hamiltonian properties of graphs and upper bounds for the general inverse degree with \( \alpha \geq 1 \).