Symmetry

Chiral interactions pose significant challenges for regularization due to the ${\gamma }_5$ Dirac matrix, which is intrinsically four-dimensional. Dimensional regularizations, while widely employed in gauge theories, encounter challenges when treating ${\gamma }_5$ in $d\ne 4$ dimensions, potentially leading to violations of chiral symmetry and the emergence of spurious anomalies. In this work, we examine aspects of Implicit Regularization, a framework formulated to operate in the physical dimension, thereby potentially avoiding ambiguities associated with ${\gamma }_5$. We discuss its implementation and implications for symmetry preservation in chiral gauge theories.

In this study, we applied dynamical system theory to analyze evolutionary trends of the higher education system, consisting of three indicators: university scale development, government financial investment, and student fee standards. The higher education system has significant uncertainty over a long period of evolution, especially with the development scale and speed of higher education largely depending on government financial investment and changes in student fee standards. We adopted the theory and methods of dynamical system theory to analyze the evolutionary trend of the higher education system, composed of these three indicators. Using the principles and methods of differential dynamics, we put forward a three-dimensional dynamic system model, involving variables such as higher education scale expansion, standard of tuition, and government financial investment. Based on the qualitative theory of ordinary differential equations, we have obtained the stability conditions for the equilibrium of the dynamic model and the conclusion of the asymptotic of the asymmetric solution of the three-dimensional dynamical system. The general integral expression of the system was obtained under specific conditions. The general integral can explain the global structure of the system in some aspects, and can compensate for the shortcomings of the local structure of equilibrium points.

Energetic powder processing includes comminution, sieving, drying, conveying, mixing, and packaging, all of which determine product performance and safety. With growing requirements for efficiency and reliability, numerical simulation has become essential for analyzing mechanisms, optimizing parameters, and supporting equipment design. This review summarizes recent progress in simulation techniques such as the discrete element method (DEM), computational fluid dynamics (CFD), and multi-scale coupling while also evaluating their predictive capabilities and limitations across various unit operations and safety concerns such as electrostatic hazards. It, thus, establishes the core “property–parameter–performance” relationships and clarifies mechanisms in multiphase flow, energy transfer, and charge accumulation, and highlights the role of symmetry in improving simulation efficiency. By highlighting persistent challenges, this work lays a foundation for future research, guiding the development of theoretical frameworks and practical solutions for advanced powder processing.