Section 1 of this study comprises the introduction. In Section 2, we present a brief yet concise history of the fundamentals of Zeta Correspondence. While we did not delve deeply into definitions, our aim is to give the reader a sense of the breadth of research surrounding Zeta Correspondence. Section 3, turning from “expansion,” explains the “heart,” the essence of Zeta Correspondence. Section 4 carefully explains Zeta Correspondence in a cycle (one-dimensional torus), which initiates the research. Section 5 deals with the Konno–Sato theorem, which is the key to discovering Zeta Correspondence. Section 6 concludes the study by discussing some Zeta Correspondence-related topics.

Hyper-dual numbers (HDNs) are defined using distinct nilpotent elements. Extending functions from real space to hyper-dual space enables the calculation of high-order derivatives without relying on formulas such as the Leibniz rule or the chain rule. Furthermore, the HDN framework facilitates computing high-order derivatives for implicit functions, such as eigenvalues. This paper presents a comprehensive overview of our research on HDN theory and its application. We explore specific topics such as the HDN-based higher-order derivative formulation, HDN matrix representation, HDN-based numerical differentiation of eigensystems, and its application in hyperelastic–plastic materials. The development of high-accuracy numerical methods using higher-order derivatives and the efficient computing of high-order derivatives are two major benefits of HDN-based numerical differentiation.

The spontaneous emergence of patterns in cell collectives is a multidisciplinary field that lies at the intersection of nonequilibrium physics and cellular biology, which sheds light on intricate biological phenomena. Despite considerable efforts, the precise integration of mechanical and chemical signaling, which drive these collective behaviors, remains elusive at the cellular level. In this study, we highlight the conserved occurrence of spatiotemporal waves involving both cell density and extracellular signal-regulated kinase (ERK) activation, observed in vitro and in vivo. We demonstrate that intercellular coupling, mediated by ERK-driven mechanochemical feedback, enables guidance cues to traverse over long distances. We further show that the patterns can be quantitatively explained by the coupling between active cellular tensions and the mechanosensitive ERK pathway—supported by our proposed mathematical model and with the backing of mechanical and optogenetic perturbation experiments. This study establishes a crucial link between the biophysical origins of spatiotemporal instabilities and the fundamental design principles that govern the efficient long-range transference of biological information.