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The Stern-Brocot Diagram and Linear Recursive Sequences
No full textHow can we visualize the convergence of rational approximations to any real number? We will use continued fractions, the iterative process that generates rational approximations, and the Stern-Brocot Diagram, which defines the unique structure and relationship between rationals in lowest terms. A Farey Recursive Function (F) is defined on the points of the Stern-Brocot Diagram, where the F-values of points lying on any line in the diagram satisfy a 2-term linear recurrence relation. For example, the Fibonacci sequence on the nonnegative integers can be extended across all rational numbers by a Farey Recursive Function.
But what happens when we look at the F-values of points that lie on a Euclidean line that is not a line in the diagram?
This presentation introduces Cross Recursion, proving that sequences of F-values along these Euclidean lines not in the diagram still follow linear recurrence relations. Our main result shows that the F-values of points lying on a Euclidean line passing through (p/q,1/q) and (a/b,0) is an interleaving of φ(w) linear recursive sequences, where φ(w) is the Euler totient function and w=|pb − qa|. Each of these φ(w) sequences has an order at most w+1. Furthermore, the F-values of points lying on a horizontal line y=1/w is an interleaving of φ(w) linear recursive sequences, where each of these φ(w) sequences is a bi-infinite sequence with an order at most w+1.
Download the attached PDF to see the abstract with proper math formatting
CHALLENGED: A PRACTICE-BASED ANALYSIS OF SIMULATING THE REALITIES OF TEACHING THROUGH PLAY
No full textChallengED is a classroom simulation game designed to explore the emotional and procedural realities of teaching through interactive play. Rather than attempting to manufacture empathy, the project focuses on conveying the unpredictability, humor, and fatigue that shape daily classroom life. Drawing from my own teaching experience, the game uses experiential realism and embodied interaction to communicate how educators navigate constant interruptions, shifting demands, and limited control. The development process also became a therapeutic practice, offering a way to reinterpret professional challenges through creative experimentation. Situated within practice-based research, ChallengED demonstrates how game design can function as both artistic inquiry and a reflective tool. The project also holds potential value for pre-service teacher education by providing a low-risk environment in which players can confront the cognitive load and emotional labor of teaching. This study contributes to ongoing discussions about how serious games represent complex professions and support reflective learning
PLANT-WILDLIFE DYNAMICS IN THE TROPICAL RAINFOREST: PHENOLOGY, TROPHIC INTERACTIONS, AND HABITAT CONNECTIVITY
No full textThe tropical rainforests of Borneo are particularly distinctive for their inter-annual phenology, with supra-annual mast fruiting events that cause dramatic fluctuations in fruit availability. Tracking its phenology is therefore vital for forest wildlife, especially frugivores and granivores whose populations depend on these resources. This dissertation investigates animal–habitat dynamics in Borneo, examining it from the perspective of plant phenology, animal responses, and restoration practices.
In the first chapter, I examined phenological change in rainforests of Borneo, focusing on tracking dipterocarp tree flowering using unoccupied aerial vehicles (UAVs). Dipterocarps exhibit the supra-annual cycle, during which individual trees bloom for four to five weeks. This short duration, coupled with persistent cloud cover, makes it difficult to monitor flowering using space-borne remote sensing. I demonstrated that UAV imagery, with spatial resolutions ranging from 20 centimeters to 5 meters, can reliably distinguish flowering in individual trees, highlighting the potential of UAVs to provide large-scale, high-resolution phenological monitoring in tropical forests.
In the second chapter, I investigated how the fruiting of dipterocarps influences habitat use across trophic levels, drawing parallels to bottom-up cascades observed in temperate systems. I tested whether pulses in dipterocarp fruit availability propagate through animal communities using Structural Equation Modelling (SEM). My results revealed that resource pulses significantly influenced certain primary consumers but had little detectable effect on secondary consumers. The lack of response at higher trophic levels suggests that tropical animal communities may be buffered against cascading effects of pulsed resources, perhaps due to the presence of alternative food sources or the dominance of large-bodied granivores that mostly lack predators.
Because many tropical rainforest animals must remain mobile to track resources, maintaining and restoring landscape connectivity is essential to support their movements. In my final chapter, I conducted a systematic review of the literature on forest restoration in Malaysian Borneo to assess whether indicators reported included measures of wildlife connectivity. My analysis revealed that none of the reviewed studies incorporated landscape scale connectivity indicators, highlighting a major gap in ensuring restoration outcomes for wildlife. I outlined recommendations to improve restoration practices, especially towards achieving the Kunming–Montreal Global Biodiversity Framework Target 2
QUANTIFYING ECOSYSTEM RESILIENCE TO WILDFIRE IN THE WESTERN UNITED STATES
No full textThe broad theme of my dissertation is to move the study of ecological resilience from a qualitative or purely theoretical framework to a quantitative framework that identifies resilient and non-resilient ecosystems and landscapes by integrating the recovery of ecosystem function, structure, and composition. The three chapters of this dissertation follow a logical progression from a landscape scale, to a watershed scale, and finally to the regional scale of the western US. First, I develop and apply an integrated Ecosystem Resilience Index (ERI) to an individual fire. Second, I characterize postfire vegetation type conversion and associated changes in water and carbon fluxes in a coastal southern California watershed. Finally, I implement the locally validated ERI to more than 2,000 fires across the western US and examine how resilience varies across ecoclimatic regions. The three chapters in this dissertation contribute to a growing body of literature which addresses fire-driven ecological change in the western US and emphasize the ecological and practical importance of the spatial and temporal characterization of resilience