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Specific energy and momentum are two fundamental quantities to investigate associated water depths, i.e., subcritical and super-critical, for open channel flow passing through wide rectangular channels. Historically, alternative or sequent depths were often used because there were no analytical solutions for flow depths with clear mathematical steps. In this study, we provide primarily detailed derivation procedures of the analytic solutions for the two water depths in specific momentum and energy equations without using symbolic software. More importantly, we found reciprocal relationships between the water depths of the specific energy and momentum and generalized specific energy and momentum equations for the quasi-rectangular channel geometries with varying depths. 

The Adomian decomposition method (ADM) is a universal approach to solving governing equations in various engineering and technological applications. The applicability of the ADM is almost limitless due to its universal applicability, but its convergence rate and numerical accuracy are sensitive to the number of truncated terms in series solutions. More importantly, Adomian formalism still holds unresolved issues regarding the mismatch of the order of the expansion parameter. The current work provides an in-depth analysis of Adomian's decomposition method, Lyapunov's stability theory, and the nonlinear perturbation theory to resolve the fundamental mismatch with physical interpretation. 

A heiau is a Hawaiian pre-Christian place of worship composed of rocks and holding historical and cultural significance, with a variety of uses ranging from worship to rituals. The scientific origins of Hawaiian heiau have yet to be fully inspected in the literature. This paper covers Hāpaialiʻi heiau on the island of Hawai‘i and its main function as a seasonal calendar through continuous observation of the sunrise and sunset locations over the sea horizon. Understanding the calendar functionality of the heiau contributes an essential step towards understanding Hawaiians’ use of indigenous science. This study mathematically investigates the sacred structure’s functionality with numerical accuracy and scientific rigour. A brief overview of the history of the Kahaluʻu area and the surrounding heiau provides important context for the heiau’s significance. This study reports and analyses the heiau’s relevant history, cultural significance and reconstruction by Kamehameha Schools. The importance of the calendar lies in the determination of the responsibilities for the current season, indicating which crops are to be harvested and which fish are in the mating season. In the mathematical analysis, the coordinates of the stones within the heiau are independently measured using Google Earth and Google Maps. The relative distances and angles between the internal stones are calculated using Google Sheets and Matlab to discover the underlying sciences possibly used by ancient native Hawaiians. 

Continuous post-endovascular aneurysm repair (EVAR) surveillance is critical for patients who have undergone a stenting procedure. Despite its life-saving importance, studies have reported that patients gradually discontinue post-EVAR surveillance due to the significant burden associated with CT scans, X-rays, etc. Considering the importance and necessity of post-EVAR surveillance, we introduce a self-resonating flexible tube implanted in a standard clinically approved stent that enables wireless sensing of blood pressure inside an aneurysm through inductive coupling. To enable blood pressure monitoring outside the body, we designed and fabricated a spiral-type antenna, specifically crafted to capture the inherent resonance frequency of the self-resonating stent tube. To showcase the wireless pressure monitoring capability, a 3D-printed elastic model of an abdominal aortic aneurysm (AAA) was prepared. Our quantitative study validated the pressure-sensing capability with adequate sensitivity (up to 687.5 Hz/mmHg) when tissue was located between the stent and the external antenna. The wireless sensing also presents a consistent linear shift in resonance frequency across all tested measuring distances as the applied pressure ranges from 60 to 140 mmHg. The proposed self-resonating stent offers promising insights for improving post-EVAR surveillance.