The aim of this special issue is to provide a deep look into today's challenging mathematical and technological problems. The authors examine and present innovative and rigorous solutions to a wide range of topics in the field of applied mathematics and engineering, including quantum cryptography, fuzzy logic, physics, and more.These studies demonstrate the importance of interdisciplinary collaboration and the combination of mathematical and technological approaches in solving the most complex problems of our time. Each of the articles is an example of the dedication of researchers in seeking effective and sustainable solutions to novel problems in science.For instance, in [1], the authors develop a quantum key distribution protocol, a method for secure communication, on qudits, to deal with the problem of cryptography systems in terms of quantum computing. The authors present a solution using quantum operators to achieve a secure key distribution on these systems.In [2], the focus is on fuzzy logic, a branch of mathematics that allows for the creation of more flexible and adaptable systems that can better handle the uncertainty and complexity of real-world problems. Specifically, in this work they provide a first notion of inconsistency by means of the absence of models, and they define two measures of consistency that belong purely to the fuzzy paradigm.The articles [3,4] and [5] present important advances in mathematical methods applied to biology and physics. The first deals with the Keller-Segel model, a mathematical model used to describe the movement of biological organisms. The second presents a direct method for finding solutions to split quaternion matrix equations, which is an important mathematical equation used to model physical systems. And the last one provides a method for determining the integrability of GL(2, R) invariant fourth-order ordinary differential equations, which is crucial for understanding the behavior of physical systems modeled by these equations.Lastly, the work [6] provides a solution for controlling multi-input linear systems using Kalman reduced form and state feedback. The authors apply this solution over Hermite rings, a type of mathematical ring, to achieve a more effective control of the system.In summary, these articles are a testament to the importance of scientific research and technological innovation in seeking solutions to the most complex problems we face today.