Tropical linear algebra is the study of classical linear algebra problems with arithmetic

done over the tropical semiring, namely with addition replaced by max, and multiplication

replaced by addition. It allows one to reformulate nonlinear problems into piecewise-linear

ones. This approach has successfully been employed to solve and characterize solutions to

many problems in combinatorial optimization, control theory and game theory [5]. Tropical

spectral theory, the study of tropical eigenvalues and eigenspaces, often plays a central role

in these applications. We derive the basics of this theory in Chapter 1.

In Chapter 2 we give a combinatorial description of the cones of linearity of the tropical

eigenvector map. In Chapter 3 we extend this work to cones of linearity of the tropical

eigenspace and polytrope map. Our results contribute to a better understanding of the

polyhedral foundations of tropical linear algebra.

Chapter 4 illustrates the above results in the context of pairwise ranking. Here one as-

signs to each pair of candidates a comparison score, and the algorithm produces a cardinal

(numerically quantified) ranking of candidates. This setup is natural in sport competitions,

business and decision making. The difficulty lies in the existence of inconsistencies of the

form A > B > C > A, since pairwise comparisons are performed independently. Tropi-

calRank is an algorithm pioneered by Elsner and van den Driessche. Solution sets of this

ranking method are precisely the polytropes studied in Chapter 3. For generic input pair-

wise comparison matrices, this set contains one unique point that is the tropical eigenvector,

which is then interpreted as the comparison score. In particular, the results in Chapter 3

provide a complete classification of all possible solution sets to the optimization problem

that TropicalRank solves. This answers open questions from several papers [22, 32] in the

area.

In Chapter 4 we also show that TropicalRank belongs to the same parametrized family

of ranking methods as two other commonly used algorithms, PerronRank and HodgeRank.

Furthermore, we show that HodgeRank and PerronRank asymptotically give the same score

under certain random ranking models. Despite their mathematical connections, we can

construct instances in which these three methods produce arbitrarily different rank order.2

The last two chapters are topics in applied probability. Chapter 5 studies the exact

and asymptotic distribution of size-biased permutations of finite sequences with independent

and identically distributed (i.i.d) terms. The size-biased permutation of a positive summable

sequence (x1 , x2 , . . .) is the same sequence presented in a random order (x[1], x[2], . . .), where

x[1] is defined to be xi with probability proportional to its `size' xi ; given that x[1] is xi ,

the next term x[2] is defined to be xj for j = i with probability again proportional to its

`size' xj , and so on. Size-biased permutations model the successive sampling method in

ecology and oil discovery, where species (or oil reserves) are discovered in a random order

proportional to their abundances. In the ranking literature it is known as the Plackett-Luce

model, a parametrized family modeling ranking distributions. Size-biased permutation is

one of the building blocks of combinatorial stochastic processes, an area of probability with

applications to computer science [78]. Finite i.i.d sequence setup serves as a simple model for

successive sampling, or ranking with increasing number of items. We study the size-biased

permutation of such a sequence using two separate methods: Markov chains and induced

order statistics. By going back and forth between the two approaches, we arrive at more

general results with simplified proofs, and provide a Poisson coupling argument which leads

to an explicit formula for the asymptotic distribution of the last few terms in the size-biased

permutation.

Chapter 6 is about the binary Little-Hopfield network. This is an established computa-

tional model of neural memory storage and retrieval which can have exponential capacity

relative to the number of neurons. However, known algorithms have produced networks with

linear capacity, and it has been a long-standing open problem whether robust exponential

storage is possible. For a network with n neurons, the problem involves a linear program in

n2 variables and exponentially many constraints. We utilized the action of the symmetric

group on the neuron labels and successfully reduced the problem to a linear program in three

variables and three constraints. Thus we explicitly constructed simple networks that answer

the question affirmatively, with the best possible asymptotic robustness index. This work

calls for further research into Little-Hopfield networks and their applications to theoretical

neuroscience and computer science.

This research study embraced the call-to-action of bridging the compassion gap (C. Su�rez-Orozco, 2019) and a qualitative life history research approach to explore the (pre-, during-, and post-) migratory experiences of ten former newcomer high school students. Through in-depth interviews, this study sought to understand the stories of these ten 1.25-generation immigrants as they retrospectively narrate their lived experiences before, during, and after their immigration journeys, with a focused interest in displacement trauma and interrupted formal education, and the influences these experiences had on their academic and life trajectories. This study also documented the heartfelt advice each participant offered to current adolescent newcomer students. The findings of this study were shared in two distinct ways: Chapter Four was written in narrative vignettes and mainly in the voices of the participants as they relived their stories, while Chapter Five examined the ten interviews collectively. The major findings of this study revealed that the longer the immigration journey, the longer the time of interrupted formal education. Most of the participants minimized or channeled their traumatic experiences to serve others. After completing high school in the United States, these newcomer students still felt unprepared for college and career due to limited English language skills. These 1.25-generation immigrants were self-motivated individuals who advised current newcomer adolescent students to study English deeply and connect with others to help with the stressors of post-migration experiences. The researcher of this study recommended more intensive English Language Development (ELD) for the 1.25-generation immigrant students, a sustained and collaborative articulation between K-12 and Higher Education to support English Learners, an ongoing orientation system to support and connect high school newcomer students, and the incorporation of trauma-informed social and emotional learning (SEL) into the curriculum.

The endoplasmic reticulum (ER) is the site of folding and maturation for virtually all secreted and transmembrane proteins. When the ER folding capacity is overwhelmed, conserved signaling pathways in eukaryotes, collectively termed the unfolded protein response (UPR), sense this ER stress to reestablish proteostasis or initiate apoptosis in conditions of unresolved stress. The best studied of the three known UPR sensors, IRE1α, consists of an ER lumenal sensing domain, a single transmembrane helix followed by cytosolic kinase and ribonuclease (RNAse) effector domains. During ER stress, IRE1α activates by oligomerization, trans-autophosphorylation and subsequent allosteric RNase activation. Active IRE1α RNase initiates noncanonical splicing of its XBP1 mRNA substrate to allow the translation of active XBP1s transcription factors that in turn upregulate hundreds of genes to increase ER folding capacity and reduce folding load. In solution, IRE1α forms reversible oligomers that correlate with higher enzymatic activity. In ER-stressed cells, activated IRE1α dynamically clusters as discrete foci visible by fluorescence microscopy, which dissolves when stress is mitigated. This thesis presents three complementary approaches aimed to better understand the direct relationship between IRE1α oligomeric states and activity: 1. Characterizing the dynamic behavior of IRE1α molecules in mammalian cells during different UPR phases and corresponding IRE1α activation stages, 2. Solving the in situ structure of large IRE1α oligomers observed in stressed cells, and 3. Dissecting elements controlling IRE1α oligomerization. We discovered that: 1. Cellular IRE1α foci are entities with complex morphologies and dynamic behaviors consisting of two distinct populations of IRE1α: a diffusionally constrained core and a freely exchanging periphery. 2. In cells, IRE1α oligomers form ordered double helices contained in a newly described ER structure comprising extremely narrow (~28 nm) anastomosing membrane tubes. The arrangement of IRE1α molecules in this ER subdomain creates a positive feedback loop that sustains IRE1α activation and provides the structural basis for the two dynamically distinct populations observed. 3. A flexible linker region connecting IRE1α’s transmembrane helix to the effector kinase/RNase domains plays a major role in controlling IRE1α kinase/RNase domains activation and oligomerization in cells. Mutants carrying deletions or mutations within this region fail to form foci during stress and cannot mount an effective UPR. Taken together, this work bridges previous observations of IRE1α behaviors observed in vitro and IRE1α’s dynamic properties in cells to provide a more complete picture of why IRE1α oligomerization and activation are intricately linked.

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