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.

The stretched Littlewood-Richardson coefficient \(c^{t\nu}_{t\lambda,t\mu}\) was conjectured by King, Tollu, and Toumazet to be a polynomial function in \(t\). It was shown to be true by Derksen and Weyman using semi-invariants of quivers. Later, Rassart used Steinberg’s formula, the hive conditions, and the Kostant partition function to show a stronger result that \(c^{\nu}_{\lambda,\mu}\) is indeed a polynomial in variables \(\nu, \lambda, \mu\) provided they lie in certain polyhedral cones. Motivated by Rassart’s approach, we give a short alternative proof of the polynomiality of \(c^{t\nu}_{t\lambda,t\mu}\) using Steinberg’s formula and a simple argument about the chamber complex of the Kostant partition function.

In this work, we study type B set partitions for a given specific positive integer \(k\) defined over \(\langle n \rangle = \{-n, -(n-1), \cdots, -1, 0, 1, \cdots, n-1, n\}\). We found a few generating functions of type B analogues for some of the set partition statistics defined by Wachs, White and Steingrímsson for partitions over positive integers \([n] = \{1, 2, \cdots, n\}\), both for standard and ordered set partitions respectively. We extended the idea of restricted growth functions utilized by Wachs and White for set partitions over \([n]\), in the scenario of \(\langle n \rangle\) and called the analogue as Signed Restricted Growth Function (SRGF). We discussed analogues of major index for type B partitions in terms of SRGF. We found an analogue of Foata bijection and reduced matrix for type B set partitions as done by Sagan for set partitions of \([n]\) with specific number of blocks \(k\). We conclude with some open questions regarding the type B analogue of some well known results already done in case of set partitions of \([n]\).

Suppose that \(\phi\) is a proper edge-\(k\)-coloring of the graph \(G\). For a vertex \(v \in V(G)\), let \(C_\phi(v)\) denote the set of colors assigned to the edges incident with \(v\). The proper edge-\(k\)-coloring \(\phi\) of \(G\) is strict neighbor-distinguishing if for any adjacent vertices \(u\) and \(v\), \(C_\phi(u) \varsubsetneq C_\phi(v)\) and \(C_\phi(v) \varsubsetneq C_\phi(u)\). The strict neighbor-distinguishing index, denoted \(\chi’_{snd}(G)\), is the minimum integer \(k\) such that \(G\) has a strict neighbor-distinguishing edge-\(k\)-coloring. In this paper we prove that if \(G\) is a simple graph with maximum degree five, then \(\chi’_{snd}(G) \leq 12\).

Let \(2 \le k \in \mathbb{Z}\). A total coloring of a \(k\)-regular simple graph via \(k+1\) colors is an efficient total coloring if each color yields an efficient dominating set, where the efficient domination condition applies to the restriction of each color class to the vertex set. In this work, focus is set upon graphs of girth \(k+1\). Efficient total colorings of finite connected simple cubic graphs of girth 4 are constructed starting at the 3-cube. It is conjectured that all of them are obtained by means of four basic operations. In contrast, the Robertson 19-vertex \((4,5)\)-cage, the alternate union \(Pet^k\) of a (Hamilton) \(10k\)-cycle with \(k\) pentagon and \(k\)-pentagram 5-cycles, for \(k > 1\) not divisible by 5, and its double cover \(Dod^k\), contain TCs that are nonefficient. Applications to partitions into 3-paths and 3-stars are given.

Using generating functions, we are proposing a unified approach to produce explicit formulas, which count the number of nodes in Smolyak grids based on various univariate quadrature or interpolation rules. Our approach yields, for instance, a new formula for the cardinality of a Smolyak grid, which is based on Chebyshev nodes of the first kind and it allows to recover certain counting-formulas previously found by Bungartz-Griebel, Kaarnioja, Müller-Gronbach, Novak-Ritter and Ullrich.

Topological indices have become an essential tool to investigate theoretical and practical problems in various scientific areas. In chemical graph theory, a significant research work, which is associated with the topological indices, is to deduce the ideal bounds and relationships between known topological indices. Mathematical development of the novel topological index is valid only if the topological index shows a good correlation with the physico-chemical properties of chemical compounds. In this article, the chemical applicability of the novel GQ and QG indices is calibrated over physico-chemical properties of 22 benzenoid hydrocarbons. The GQ and QG indices predict the physico-chemical properties of benzenoid hydrocarbons, significantly. Additionally, this work establishes some mathematical relationships between each of the GQ and QG indices and each of the graph invariants: size, degree sequences, maximum and minimum degrees, and some well-known degree-based topological indices of the graph.