In order to explore the Lorenz-Haken model, we will concentrate on the flow curvature technique, a recently created method based on differential geometry. This approach treats a dynamical system's trajectory curve or flow as a curve in Euclidean space. Analytical calculations may be used to determine the flow curvature, which is the trajectory curve's curvature. The flow curvature manifold, which is related to the dynamical system of any dimension, is defined by the locations where the flow curvature is null. For the slow invariant manifold of the same dynamical system, the flow curvature manifold offers an analytical equation. The slow invariant manifold equation may be discovered using the flow curvature technique without the need of any asymptotic expansions. In this study, we compute the analytical equation of the slow invariant manifold for the three-dimensional Lorenz-Haken model using the flow curvature approach for the first time. This analytical equation, together with its visual representation in phase space, makes it possible to distinguish between the slow development of trajectory curves and the rapid one, which advances our knowledge of this slow-fast domain. This study also advances the field relative to earlier similar work. Aside from that, we utilize the Darboux theorem to demonstrate the slow manifold's invariance characteristic.

In order to explore the underlying reasons for the decline in birth rate, this article selects 12 explanatory variables and uses ridge regression method to study the birth rate in Henan Province from 2015 to 2021. Research has shown that four factors, namely the average salary of urban unit employees, the urbanization degree, the ratio of female employees with a university degree or above, and the population mortality rate, can not explain the birth rate. However, the proportion of gross domestic product of the second and third industries, as well as the proportion of female population over 15 years of age who are illiterate, has a positive impact on the car success rate. The gross domestic product per capita, the number of beds per 10000 people in medical institutions, the per capita disposable income of urban residents, the per capita disposable income of rural residents, the adolescent dependency ratio, and the elderly dependency ratio have a negative impact on the birth rate. Through the research in this article, the main factors affecting the birth rate in Henan Province have been identified, and policy recommendations for improving the birth rate have been proposed. The positive impact represents increasing investment in these factors, which can effectively improve the birth rate in Henan Province and solve the serious problems we are currently facing. The negative factor is the opposite.

In this article, an attempt has been made to explain and model the Taylor table method in Python. A step-by-step algorithm has been developed, and the methodology has been presented for programming. The developed TT_method() function has been tested with the help of four problems, and accurate results have been obtained. The developed function can handle any number of stencils and is capable of producing the results instantaneously. This will eliminate the task of hand calculations and the use can directly focus on the problem solving rather than working hours to descretize the problem.

Based on the relevant data of economic development of 31 provinces in China from 2016 to 2021, this paper selects the level of economic development as the response variable. The explanatory variables are selected from the aspects of location, human capital, industrial structure, foreign trade, system, science and technology and employment level. Aiming at the data with hierarchical structure, a multilevel model is proposed for analysis. It is concluded that human capital, industrial structure, foreign trade and system have a significant impact on the level of economic development. The impact of technology and employment level on the level of economic development is not significant. At the same time, this paper also makes a comparative analysis of the linear regression model, random intercept model and random intercept-slope growth model, which shows the advantages of random intercept-slope growth model.

The process of making decisions on software architecture is the greatest significance for the achievement of a software system's success. Software architecture establishes the framework of the system, specifies its characteristics, and has significant and major effects across the whole life cycle of the system. The complicated characteristics of the software development context and the significance of the problem have caused the research community to build various methodologies focused on supporting software architects to improve their decision-making abilities. With these efforts, the implementation of such systematic methodologies looks to be somewhat constrained in practical application. Moreover, the decision-makers must overcome unexpected difficulties due to the varying software development processes that propose distinct approaches for architecture design. The understanding of these design approaches helps to develop the architectural design framework. In the area of software architecture, a significant change has occurred wherein the focus has shifted from primarily identifying the result of the architecting process, which was primarily expressed through the representation of components and connectors, to the documentation of architectural design decisions and the underlying reasoning behind them. This shift finally concludes in the creation of an architectural design framework. So, a correct decision- making approach is needed to design the software architecture. The present study analyzes the design decisions and proposes a new design decision model for the software architecture. This study introduces a new approach to the decision-making model, wherein software architecture design is viewed based on specific decisions.