This paper studies the latest and state-of-the-art underwater thermal energy harvesting algorithms and techniques designed in the latest decade (2014-2024). The techniques are classified based on their unique operations for energy harvesting. This classification includes thermal energy harvesting using a phase change material (PCM), thermoelectric generator (TEG) and multi-source harvesting. Every class of techniques is described by its operation using a schematic diagram and a mathematical model to fully understand its working principle. Moreover, every individual technique is also described in terms of its operation, amount of harvested energy/power and the aspect(s) where margin of further improvement exists. Also, a comparative analysis of the classified algorithms is performed with each other as well as with other underwater energy harvesting techniques (solar, piezoelectric, wave) to highlight their effectiveness and feasibility in a diverse set of underwater and various other applications. The classified techniques are also compared in terms of harvested output to indicate their harvesting efficiency. Furthermore, the publications made in the latest decade in terms of thermal energy harvesting using PCM, TEG and multi-source methods are also graphically depicted. Such a description of the studied techniques and classified methods is unique from the already existing underwater energy harvesting reviews in literature where an in-depth and thorough analysis is absent, rather only marginal description is given. The harvesting results indicate that hybrid (multi-source) and PCM methods have the greatest amount of harvested power and energy, respectively. Finally, the research challenges in underwater thermal energy harvesting are specified and areas of further research are highlighted for future investigation.

This paper presents new results on the analysis of positive linear time-invariant systems with non-negative state variables and output data for non-negative initial conditions and inputs. The positive linear time-invariant systems are characterized by the state-space equations which offer a formal mathematical structure of form $$\frac{dx(t)}{dt} = Ax(t)$$ where the system matrix $A\in \mathbb{R}^{n,n}$ is Metzler, with non-negative off-diagonal components. These systems exhibit essential characteristics such as monotonicity, stability, and non-negativity, making them fundamental in applications such as biological systems, mathematical economics, chemical reaction networks, and transportation models. We present the theoretical foundations utilizing a mathematical framework from algebraic systems, matrix theory, and stability analysis to investigate stability, $\mathfrak{D}$-stability, and strong $\mathfrak{D}$-stability of positive linear time-invariant systems in the presence of Metzler and Hurwitz matrices. The numerical testing supports the spectrum analysis and $\epsilon$-pseudospectrum of Metzler matrices.

Several recent papers gave numerical radius inequalities for sums and products of operators which are defined on a complex separable Hilbert space H. In this paper, we prove a numerical radius inequality which generalizes and refines a recent inequality proved by Kittaneh.

In this paper, we give some basic results concerning Toeplitz operators whose symbol is of the form f(θ)ϕ, where ϕ is a radial function and f(θ) is a polynomial in eiθ, then use these results to characterize all Toeplitz operators which commute with them on the Bergman space.