This thesis is a supplement textbook designed with ASU’s MAT 370, or more generally, a course in introductory real analysis (IRA). With research in the realms of mathematics textbook creation and IRA pedagogy, this supplement aims to provide students or interested readers an additional presentation of the materials. Topics discussed include the real number system, some topology of the real line, sequences of real numbers, continuity, differentiation, integration, and the Fundamental Theorem of Calculus. Special emphasis was placed on worked examples of proven results and exercises with hints at the end of every chapter. In this respect, this supplement aims to be both versatile and self-contained for the different mathematics skill levels of readers.

This thesis project focuses on algorithms that generate good sampling points for function approximation. In one dimension, polynomial interpolation using equispaced points is unstable, with high Oscillations near the endpoints of the interpolated interval. On the other hand, Chebyshev nodes provide both stable and highly accurate points for polynomial interpolation. In higher dimensions, optimal sampling points are unknown. This project addresses this problem by finding algorithms that are robust in various domains for polynomial interpolation and least-squares. To measure the quality of the nodes produced by said algorithms, the Lebesgue constant will be used. In the algorithms, a number of numerical techniques will be used, such as the Gram-Schmidt process and the pivoted-QR process. In addition, concepts such as node density and greedy algorithms will be explored.

Pelvic Circumferential Compression Devices (PCCDs), an important medical device when caring for patients with pelvic fractures, play a crucial role in the stabilization and reduction of the fracture. During pelvic fracture cases, control of internal bleeding through access to the femoral artery is of utmost importance. Current designs of PCCDs do not allow vital access to this artery and in attempts to gain access, medical professionals and emergency care providers choose to cut into the PCCDs or place them in suboptimal positions with unknown downstream effects. We researched the effects on surface pressure and the overall pressure distribution created by the PCCDs when they are modified or placed incorrectly on the patient. In addition, we investigated the effects of those misuses on pelvic fracture reduction, a key parameter in stabilizing the patient during critical care. We hypothesized that incorrectly placing or modifying the PCCD will result in increased surface pressure and decreased fracture reduction. Our mannequin studies show that for SAM Sling and T-POD, surface pressure increases if a PCCD is incorrectly placed or modified, in support of our hypothesis. However, opposite results occurred for the Pelvic Binder, where the correctly placed PCCD had higher surface pressure when compared to the incorrectly placed or modified PCCD. Additionally, pressure distribution was significantly affected by the modification of the PCCDs. The cadaver lab measurements show that modifying or incorrectly placing the PCCDs significantly limits their ability to reduce the pelvic fracture. These results suggest that while modifying or incorrectly placing PCCDs allows access to the femoral artery, there are potentially dangerous effects to the patient including increased surface pressures and limited fracture reduction.