Urban Transportation Network Design (TND) is a highly complex system engineering. It has always been regarded as a very difficult problem in urban planning and transportation. This study tries to sort out and summarize the existing research system and development process in this field. From the perspective of mathematical modeling, the multi-objective property, nonlinear characteristics, uncertainty and NP-hard computational complexity of the TND problem are explained one by one, and a series of solution difficulties derived from them also appear. Classical optimization methods, such as linear programming, integer programming, bi-level programming models and so on, have advantages in solving structured problems. But their limitations in large-scale dynamic situations are also pointed out. In comparison, new intelligent and simulation optimization methods, such as heuristic algorithms, data-driven modeling strategies and simulation-optimization coupling frameworks, provide more potential ways to solve complex traffic network design problems. The current mainstream research paradigm can be summarized as the progressive process of "theoretical analysis—optimization solution—simulation verification". Different tools work together in this process to deepen the understanding of the problem and improve the solution efficiency. Network planning with dynamic and multiple uncertainties still faces many unsolved difficulties. Big data technology, artificial intelligence methods and the interdisciplinary integration may become the key driving forces for the future breakthrough in the TND field. This review aims to provide basic theoretical reference and framework support for the follow-up related research.

Blood oxygen saturation is a critical physiological parameter for evaluating human cardiopulmonary function and blood oxygen-carrying capacity. Non-invasive pulse oximeters based on the principle of photoplethysmography (PPG) have been widely used in clinical monitoring and home health care due to their simple operation and rapid response. This paper reviews the core technologies of pulse oximeters, focusing on the operating principles of their photoelectric sensor modules, hardware circuit design, key performance indicators, and the challenges they face. The article discusses in detail the differences between transmissive and reflective sensors, as well as the design of key circuits in signal processing such as amplification, filtering, and analog-to-digital conversion. Furthermore, it points out the impact of factors such as motion artifacts, low perfusion, and skin color differences on measurement accuracy. Finally, the future development of oximeters towards intelligence, multi-functionality, and high robustness is prospected. This paper aims to provide a theoretical reference for the design optimization and performance improvement of oximeters.

This study used information entropy as the core mathematical tool and focused on six typical global cultural scales: the Chinese pentatonic scale, the Indian 22-shruti scale, the Arabic 17 tone and other rhythmic scales, the European natural major scale, the Japanese mode scale, and the Indian pentatonic scale. By quantifying the uncertainty of interval distribution, the mathematical laws behind different cultural music aesthetics have been revealed. This study first constructed a frequency system for each scale and conducted interval statistics. Then, the Shannon entropy formula was used to calculate the entropy value, and the entire process was automatically calculated through Python programming. Finally, the deep logic of entropy value differences was explained in combination with cultural background. The results indicate that entropy is highly correlated with the number of core notes and the diversity of intervals in the scale: low entropy scales (2.0-2.8 bits) correspond to the cultural aesthetics of regular harmony, while high entropy scales (3.1-3.7 bits) relate to improvisation and diverse music styles, providing a quantitative analysis framework for cross-cultural music research.

In classical Newtonian theory, space and time are treated as separate entities, where time flows independently and space is Euclidean. However, this framework becomes inadequate in describing phenomena involving light and strong gravitational fields. This limitation motivates the development of general relativity, in which space and time are unified into a single geometric structure—spacetime, and gravity is interpreted as its curvature. Within this framework, the principles of geometry and symmetry determine the form of physical laws and their observable consequences. In this work, literature analysis and theoretical derivation methods are employed to study two classical observational tests of general relativity—gravitational redshift and light deflection—from a geometric and symmetry-based perspective, along with comparison to Newtonian theory. The results show that both phenomena arise naturally from the geometric structure of spacetime and can be described within a unified framework. Spacetime symmetries lead to conserved quantities, which provide an effective method for analyzing photon trajectories. The comparison with Newtonian theory highlights that the differences in predictions originate from the underlying geometric structure of spacetime.

The adjoint relation of linear operators is a core structure in inner product spaces. Normal operators, due to their commutativity with their adjoints, possess elegant spectral properties. Classical theory shows that a normal operator and its adjoint share the same eigenvectors with conjugate eigenvalues. However, this conclusion is confined to the normal case and typically relies on the spectral theorem or triangularization techniques. This paper generalizes the idea of spectral symmetry to arbitrary diagonalizable operators and reveals its essence from a purely geometric perspective. For a diagonalizable operator on a finite-dimensional complex inner product space, we construct a new operator based on its eigenspace decomposition, which acts as the conjugate of the eigenvalues on each eigenspace. Furthermore, by constructing a new inner product that depends on the eigenspace decomposition, the original operator and this new operator become mutual adjoints under this new inner product, and their relationship with the standard adjoint is established via a similarity transformation. The positive self-adjoint operator introduced herein captures the geometric inclination of the eigenspaces with respect to the original inner product. When the operator is normal, this construction reduces to the classical case. This paper provides a unified geometric explanation of the connection between the spectral structure of diagonalizable operators and the adjoint relation, offering a new perspective for operator theory and matrix analysis.