Mathematical models across the applied sciences often utilize a standard methodological representation called a state variable formulation more commonly referred to as state space form. Recent research in unmanned underwater vehicle motor turbine blade thermal modeling for fatigue-life is generalized here permitting the proposed novel state space from to be applied to electrodynamics, motion mechanics, and many other disciplines. Proposed here is a very compact form inherently representing time variance, with a convenient presentation of dynamic variables applicable to all proper transfer functions, where all the distinct, real poles, zeros and gain of the transfer function appear as explicit components in the state space. The resulting manifestation simplifies utilization of the state space methods broadly across the applied sciences.

The realization of value in eco-products represents a rapidly advancing research theme. However, the clarity of relevant concepts, types and operational mechanisms remains elusive. A comprehensive review of current-stage research outcomes in eco-product value realization not only provides a novel approach to rejuvenating regional economies but also proactively contributes to the enhancement of local ecological environments. This article conducts a systematic literature review on the fundamental concepts, research scope and pathways for realizing the value of eco-products. The conclusions are as follows: (1) To clarify the basic concepts of ecosystem services and eco-products, the interrelationship between the two and the research scope of eco-products. (2) To elaborate the method for accounting the value of eco-products. (3) To demonstrate governmental path, market path and collaborative path for realizing the value of eco-products. (4)To review cases pertaining to the realization of value in public eco-products, quasi-public eco-products and operational eco-products both domestically and internationally. Building on this research foundation, the article identifies shortcomings in past research results and proposes key directions for future research progress.

Based on some geometrical properties of the Rabinovich system the closed-form solutions of the equations has been established. Moreover the Rabinovich system is reduced to a nonlinear differential equation depending on an auxiliary unknown function. The approximate analytical solutions are built using the Optimal Auxiliary Functions Method (OAFM). A good agreement between the analytical and corresponding numerical results has been performed. The accuracy of the obtained results emphasizes that this procedure could be successfully applied for more dynamical systems with these geometrical properties.

This work emphasizes some geometrical properties of the Maxwell--Bloch equations. Based on these properties the closed-form solutions of their equations are established. Thus, the Maxwell-Bloch equations are reduced to a nonlinear differential equation depending on an auxiliary unknown function. The approximate analytical solutions were built using the Optimal Homotopy Asymptotic Method (OHAM). A good agreement between the analytical and corresponding numerical results was found. The accuracy of the obtained results is validated through the representative figures. This procedure could be successfully applied for more dynamical systems with geometrical properties.