Papers / Proofs / Notes

Mathematics

Formal papers, mathematical writing, proofs, and selected explorations — presented in their original PDF form.

Labor Market Signals and College Major Choice

Differences across Institution Tiers and Income Groups

This study asks whether labor-market signals shape college major completion patterns and whether responsiveness differs across institutions and family-income groups. We combine IPEDS bachelor’s-completion data for 2014–2024 with BLS Occupational Employment and Wage Statistics, official CIP–SOC crosswalks, the 2025 Carnegie Classification, and public NPSAS PowerStats estimates for 2016 and 2020. A standardized MarketSignal combines occupational wage levels, wage growth, and employment growth and is matched to majors with a four-year lag. Model 1 uses school, major, and year fixed effects. Models 2a and 2b use grouped multinomial logit to compare institution tiers and income quartiles. The average Model 1 association is positive but small and marginally significant. R1/R2 institutions show a more positive response than other institutions, although absolute estimates are sensitive to controls. Income-quartile responses are strongly heterogeneous, with the top quartile showing the most positive response. These estimates are associative and preliminary; they identify differential responsiveness rather than proving that labor-market information causes educational inequality.

Applied Math
Numerical Analysis
Social Science

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Min-Max Degree Optimization in Edge-Minimal Hall-Type Key Assignments

This paper studies a key distribution problem on bipartite graphs G = (M, R, E), where M is a set of n members and R is a set of m rooms (m ≤ n). Specifically, we consider a system where any subset of m members must be able to simultaneously access all m rooms with distinct keys. By Hall’s Marriage Theorem, the minimum total number of keys needed to make m-room matchings is m(n − m + 1). We focus on how to distribute these keys fairly among members so that the maximum key holding number is optimized. This parameter is introduced as ∆(m, n). We first prove that every room must hold exactly n − m + 1 keys in any minimal valid assignment. Then, we demonstrate that ∆(m, n) = Ceil(m(n−m+1)/n) for all positive integers m ≤ n. Finally, we propose two future research problems on member regularity and non-abelian finite groups.

Graph Theory

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Paper Oct 2025

Exploring Volumes Bounded by Implicit Surfaces: A Visual and Multivariate Calculus-Based Approach

A discussion of the volumes of an assortment of 3D shapes defined by implicit surfaces, using both analytical and numerical methods. We begin with foundational concepts in Multivariable Calculus including triple integrals, Jacobians, etc, laying the groundwork for exact volumes of classical shapes, such as spheres, ellipsoids and torus. For more complex surfaces, Monte Carlo estimation, an useful method for estimating volumes, is employed. We then explore parameter variations to demonstrate the efficiency and versatility of combining exact integration with computational methods.

Multivariable Calculus

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