Optimal foraging theory provides a suite of tools that model the best way that an animal will <br/>structure its searching and processing decisions in uncertain environments. It has been <br/>successful characterizing real patterns of animal decision making, thereby providing insights<br/>into why animals behave the way they do. However, it does not speak to how animals make<br/>decisions that tend to be adaptive. Using simulation studies, prior work has shown empirically<br/>that a simple decision-making heuristic tends to produce prey-choice behaviors that, on <br/>average, match the predicted behaviors of optimal foraging theory. That heuristic chooses<br/>to spend time processing an encountered prey item if that prey item's marginal rate of<br/>caloric gain (in calories per unit of processing time) is greater than the forager's<br/>current long-term rate of accumulated caloric gain (in calories per unit of total searching<br/>and processing time). Although this heuristic may seem intuitive, a rigorous mathematical<br/>argument for why it tends to produce the theorized optimal foraging theory behavior has<br/>not been developed. In this thesis, an analytical argument is given for why this<br/>simple decision-making heuristic is expected to realize the optimal performance<br/>predicted by optimal foraging theory. This theoretical guarantee not only provides support<br/>for why such a heuristic might be favored by natural selection, but it also provides<br/>support for why such a heuristic might a reliable tool for decision-making in autonomous<br/>engineered agents moving through theatres of uncertain rewards. Ultimately, this simple<br/>decision-making heuristic may provide a recipe for reinforcement learning in small robots<br/>with little computational capabilities.

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.

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.