Emerging Trends in Symmetry

Symmetry, long recognized as a cornerstone of the natural sciences, has increasingly found relevance across a variety of disciplines, from physics and mathematics to economics, architecture, and systems theory. This interdisciplinary review explores the expanding role of symmetry as a conceptual and analytical tool, highlighting its applications in diverse fields. In classical and quantum physics, symmetry principles form the foundation for conservation laws, particle interactions, and field equations. In economics and decision sciences, symmetry underlies equilibrium models and optimization techniques. In design and architecture, symmetry informs aesthetics and structural integrity.

Topological insulators and superconductors constitute a class of quantum materials characterized by insulating bulks and symmetry-protected conducting boundaries. Symmetry principles, notably time-reversal (TRS), particle-hole (PHS), and chiral symmetry, play a fundamental role in determining the topological phases and their classification within the Altland-Zirnbauer scheme.

Symmetry breaking serves as a central organizing principle in the understanding of nonlinear systems across physics, biology, chemistry, and engineering.

This systematic review comprehensively examines the burgeoning field of Digital Twin (DT) technology, analyzing its current state, diverse applications, and future trajectory. Through a rigorous methodology involving a systematic search of academic databases and relevant industry literature, this review synthesizes findings across a spectrum of disciplines.

In the era of intelligent technologies and data-driven systems, multi-objective decision-making (MODM) has become an essential aspect of managing complex environments such as smart cities, autonomous systems, and cyber-physical networks.

In this study we investigate how differential equations play a role in quantum mechanics, focusing mainly on the role of a prominent one, Schrödinger's equation, which governs the behavior of quantum systems.