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  1. We consider the rank of a class of sparse Boolean matrices of size $n \times n$. In particular, we show that the probability that such a matrix has full rank, and is thus invertible, is a positive constant with value about $0.2574$ for large $n$. The matrices arise as the vertex-edge incidence matrix of 1-out 3-uniform hypergraphs. The result that the null space is bounded in expectation, can be contrasted with results for the usual models of sparse Boolean matrices, based on the vertex-edge incidence matrix of random $k$-uniform hypergraphs. For this latter model, the expected co-rank is linear in the number of vertices $n$, \cite{ACO}, \cite{CFP}. For fields of higher order, the co-rank is typically Poisson distributed. 
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  2. This project uses an ecological belonging intervention approach [1] that requires one-class or one- recitation/discussion session to implement and has been shown to erase long-standing equity gaps in achievement in introductory STEM courses. However, given the wide social and cultural heterogeneity across US university contexts (e.g., differences in regional demographics, history, political climates), it is an open question if and how the intervention may scale. This project brings together an interdisciplinary team across three strategically selected universities to design, test, and iteratively improve an approach to systematically identify which first and second year courses would most benefit from the intervention, reveal student concerns that may be specific to that course, adapt the intervention to address those concerns, and evaluate the universality versus specificity of the intervention across university contexts. This systematic approach also includes persuasion and training processes for onboarding the instructors of the targeted courses. The instructor onboarding and the intervention adaptation processes are guided by a theory-of-action that is the backbone of the project’s research activities and iterative process improvement. A synergistic mixture of qualitative and quantitative methods is used throughout the study. In this paper, we describe our theoretical framing of this ecological belonging intervention and the current efforts of the project in developing customized student stories for the intervention. We have conducted focus groups across each of the partner institutions (University of Pittsburgh, Purdue University, and University of California Irvine). We describe the process of developing these contextually relevant stories and the lessons learned about how this ecological belonging intervention can be translated across institutional contexts and for various STEM majors and systemically minoritized populations. The results of this work can provide actionable strategies for reducing equity gaps in students' degree attainment and achievement in engineering. 
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  3. Let ${\bf A}$ be an $n\times m$ matrix over $\mathbf{GF}_2$ where each column consists of $k$ ones, and let $M$ be an arbitrary fixed binary matroid. The matroid growth rate theorem implies that there is a constant $C_M$ such that $m\geq C_M n^2$ implies that the binary matroid induced by {\bf A} contains $M$ as a minor. We prove that if the columns of ${\bf A}={\bf A}_{n,m,k}$ are chosen \emph{randomly}, then there are constants $k_M, L_M$ such that $k\geq k_M$ and $m\geq L_M n$ implies that ${\bf A}$ contains $M$ as a minor w.h.p. 
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  4. Abstract Prominent scarps on Pinedale glacial surfaces along the eastern base of the Teton Range confirm latest Pleistocene to Holocene surface‐faulting earthquakes on the Teton fault, but the timing of these events is only broadly constrained by a single previous paleoseismic study. We excavated two trenches at the Leigh Lake site near the center of the Teton fault to address open questions about earthquake timing and rupture length. Structural and stratigraphic evidence indicates two surface‐faulting earthquakes at the site that postdate deglacial sediments dated by radiocarbon and optically stimulated luminescence to ∼10–11  ka. Earthquake LL2 occurred at ∼10.0  ka (9.7–10.4 ka; 95% confidence range) and LL1 at ∼5.9  ka (4.8–7.1 ka; 95%). LL2 predates an earthquake at ∼8  ka identified in the previous paleoseismic investigation at Granite Canyon. LL1 corresponds to the most recent Granite Canyon earthquake at ∼4.7–7.9  ka (95% confidence range). Our results are consistent with the previously documented long‐elapsed time since the most recent Teton fault rupture and expand the fault’s earthquake history into the early Holocene. 
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  5. We analyze the cover time of a biased random walk on the random graph G_{n,p}$ The walk is biased towards visiting vertices of low degree and this makes the cover time less than in the unbiased case. 
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