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Full-Text Articles in Physical Sciences and Mathematics
The Zariski-Riemann Space As A Universal Model For The Birational Geometry Of A Function Field, Giovan Battista Pignatti Morano Di Custoza
The Zariski-Riemann Space As A Universal Model For The Birational Geometry Of A Function Field, Giovan Battista Pignatti Morano Di Custoza
Dissertations, Theses, and Capstone Projects
Given a function field $K$ over an algebraically closed field $k$, we propose to use the Zariski-Riemann space $\ZR (K/k)$ of valuation rings as a universal model that governs the birational geometry of the field extension $K/k$. More specifically, we find an exact correspondence between ad-hoc collections of open subsets of $\ZR (K/k)$ ordered by quasi-refinements and the category of normal models of $K/k$ with morphisms the birational maps. We then introduce suitable Grothendieck topologies and we develop a sheaf theory on $\ZR (K/k)$ which induces, locally at once, the sheaf theory of each normal model. Conversely, given a sheaf …
Prime Factors: America’S Prioritization Of Literacy Over Numeracy And Its Relationship To Systemic Inequity, Troy Smith
Prime Factors: America’S Prioritization Of Literacy Over Numeracy And Its Relationship To Systemic Inequity, Troy Smith
Dissertations, Theses, and Capstone Projects
For much of American history, literacy has been prioritized in K-12 education and society, at large, at the expense of numeracy. This lack of numerical emphasis has established innumeracy as an American cultural norm that has resulted in America not producing a sufficient number of numerate citizens, and ranking poorly on mathematical performance in international comparisons. This paper investigates the decisions and circumstances that led to this under prioritization, along with the public and cultural impact of said actions. Toward this end, literature regarding contemporary and historical influences on American mathematics education (e.g., civic, policy, and parental) was reviewed. The …