Based on the definition of volatility and conditional value risk (CVaR), this paper introduces the implied volatility into CVaR model, and further analyzes the partial differential equation of stock portfolio optimization in the form of BS model. In the process of multi-stage investment, in order to reasonably control the investment risk of each stage, the CvaR model based on implied volatility is constructed by using the scenario tree method. With the data of 1166 trading days as the data, 4 stock assets as the data set of this study, the optimization model is applied to the calculation and analysis. The numerical simulation shows that the stock price fluctuation of the four multi-cycle stocks ranges from -23.45% to 41.97%, showing a clustering phenomenon. Among them, the volatility of stocks A and C is more obvious than that of stocks B and D, and the probability density tails of stocks are longer in the cycle, and they all show thick tail characteristics, indicating that the introduction of implied volatility of CVaR model makes the risk control of actual equity asset investment more reasonable.

In this paper, the characteristics and distribution of the spatial clustering diffusion characteristics and distribution of the spatial accumulation of rural areas are quantified by using the GIS space analysis method, the analysis method of the nuclear density estimation, the hotspot analysis, the spatial self-correlation, and the large number of the rural areas of Chongqing. Compared with the difference of the amount of the education facility in Chongqing, the difference between the amount of the education facility was compared, and the development gap of the education facility was assessed. The study showed that in 2023, the imbalance coefficient of the school of compulsory education in Chongqing was reduced from 0.3637 in 2013 to 0.02433 in 2023, and the primary school stage was reduced from 0.3582 to 0.1952. This paper shows that the imbalance coefficient of education resource layout in Chongqing is decreasing year by year, and the spatial equilibrium of resource space increases. This study provides the effective thinking and method for the adjustment of the education resource space layout structure in Chongqing, and provides the scientific decision basis for the calibration of the existing planning and the formulation of future planning.

This paper discusses the appreciation of the elderly to influence the actual exchange rate by using the requirements structure and the current account mechanism. Using the internal actual exchange rate formula and the Balassa-Samuelson effect, the propagation mechanism of the aging of the population was established. This paper discusses the influence of aging on trade balance, and sets up the panel model of countries of different age categories. Through heterogeneity analysis and multivariate regression test assessment. The study of mathematical methods found that the rate of pension care significantly affected the actual effective exchange rate. In countries where aging and moderate aging lead to depreciation, aging and non-ageing countries can rise.

In this paper, the basic Wiener filter structure and adaptive algorithm module are used to optimize the parameter adjustment and data noise processing in the adaptive filter algorithm. Based on the LMS criterion, the algorithm is further refined by quantization error and affine projection optimization, which improves the accuracy and speed of vortex and circulation data analysis. The optimized algorithm reduces noise and covariance error, and achieves excellent performance in filtering evaluation (SRTAE: \(1.623\times10^{-2}\,\text{m}\) and \(1.162\times10^{-4}\,\text{m/s}\)). The results show that the spatio-temporal coupling effect between vortex and circulation can be found through numerical modeling and spatio-temporal analysis. This study provides a valuable reference for promoting the application of computational mathematics in the field of climate monitoring.

Cultural heritage represents the historical and cultural achievements of a nation, playing a vital role in studying human civilization and preserving national languages and scripts. This study utilizes virtual simulation technology to design a virtual pavilion for Chinese language and writing, employing image and text feature extraction algorithms for feature fusion and 3D modeling. The effectiveness of Chinese character extraction is validated through feature point matching, while the virtual exhibition’s impact is assessed via user experience scores. Results indicate that the proposed algorithm achieves accurate extraction with no misrecognition. User interest rankings highlight text images as the most influential factor, followed by visual imagery, pavilion experience, scene art, and language culture. Analysis of user feedback shows an average experience score exceeding 60 points, confirming the pavilion’s effectiveness in preserving and promoting Chinese language and writing culture.

In recent years, due to the adjustment of economic structure, the people’s living standard and the increase of leisure time, the sports industry has become a new economic growth point. This paper studies and analyzes the characteristics of the industry background and business background of the sports industry, explores the factors and internal driving force affecting the design of its business model, and fully analyzes the mechanism, functional role, and logical relationship of the elements for constructing the business model of the sports industry, and then explores the characteristics of the business style of the sports industry. From the perspective of knowledge state, using the reinforcement learning mechanism, the evolution process of the sports industry business model from the first stage to the fourth stage is described. Taking Company H as a research case, the process and economic effect of the transformation and upgrading of its business model through the reinforcement learning mechanism is analyzed and it is found that as of 2023 the company’s operating income has increased by 2.4 times through transformation and upgrading, and its net profit has increased by 125.57 percentage points compared to 2016. It further understands the role that the enhanced learning mechanism brings to the development of the sports industry, and expects to be able to provide a reference for the sports industry to carry out business model transformation in the future.

We initiate a study of the toughness of directed graphs by considering the natural generalization of that for ordinary graphs. After providing some general results, computations are completed for a few natural examples. Maximum possible toughness is also considered. Some open problems are  posed.

Let \(G\) and \(H\) be graphs and \(1\) be a positive number. An \(H\)-irregular labeling of \(G\) is an assignment of integers from \(1\) up to \(k\) to either vertices, edges, or both in \(G\) such that each sum of labels in a subgraph isomorphic to \(H\) are pairwise distinct. Moreover, a comb product of \(G\) and \(H\) is a construction of graph obtained by attaching several copies of \(H\) to each vertices of \(G\). Meanwhile, an edge comb product of \(G\) and \(H\) is an alternate construction where the copies of \(H\) is attached on edges of \(G\) instead. In this paper, we investigate the vertex, edge, and total \(H\)-irregular labeling of \(G\) where both \(G\) and \(H\) is either a comb product or an edge comb product of graphs.

This study applies Support Vector Machine (SVM) and K-Nearest Neighbors (KNN) algorithms to classify five types of basketball footwork. SVM maps the training data into a high-dimensional space using nonlinear transformation and classifies it with support vectors and a hyperplane. Experimental analysis showed minimal differences in peak and trough values of footwork movements; therefore, only mean and standard deviation features were retained, resulting in 12 effective features. KNN experiments demonstrated that recognition accuracy varies with different K values. The highest accuracy (80.7%) was achieved when K = 5 with the selected features. The study also examined the physical characteristics of basketball players, analyzing height, weight, and other indicators. Statistical results showed no significant body shape differences between experimental and control groups (P > 0.05). A T-test on dribbling, shooting, and layup performance also revealed no significant differences between the groups (P > 0.05).

This study explores the employment competitiveness of computer science majors by integrating combinatorial mathematics into the evaluation process. Utilizing the Analytic Hierarchy Process (AHP) and the improved FKCM clustering algorithm, we construct a hierarchical model to assess the impact of entrepreneurial education, learning motivation, and investment on job competitiveness. Data from 314 participants were analyzed using combinatorial techniques to derive optimal weightings for each factor, ensuring the evaluation model’s robustness. The results highlight significant gender differences in practical and feedback-based entrepreneurship education, with males outperforming females. However, no notable differences were observed in job interest, learning motivation, or overall employment competitiveness.