Notes
Working notes on definitions, examples, and computations.
Browse notesMathematics
Notes, papers, projects & proofs
The working archive of Drew M. Schaffner: course notes typeset in LaTeX, papers, computational projects, and proofs from graduate study, with the Python code and graphics that accompany them.
Sections
Papers
Finished papers with their sources and figures.
Browse papersProjects
Computational projects with Python code, experiments, and graphics.
Browse projectsGraduate Proofs
Proofs written during graduate study in mathematics.
Browse graduate proofsRecently published
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Note Analytic Geometry
Coordinate proofs in plane geometry, conics and their reflective properties, rotation of axes, and parametric curves.
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Note Classical Mechanics
First principles of Newtonian mechanics and the Galilean group, following Arnold.
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Note Complex Analysis
Series, singularities and residues, worked through as examples.
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Note Foundations
Statements and proof, the number systems, parity, and the irrationality of the square root of two.
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Paper Harmony Amid Chaos
Galois tables of GF(p²): three theorems on the symmetry between twin tables, and two measures, Arnold's and our own, of how randomly the logarithms are spread.
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Project Who Gets the Loan?
How much more often are Black, Hispanic and Asian mortgage applicants denied than White applicants in the 2025 HMDA data, and how much of each gap survives controls for income, debt-to-income, loan-to-value, loan type, state and lender? More than half of the Black–White gap remains, but without credit scores that is not proof of discrimination.
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Proof A Power Series with a Natural Boundary
The lacunary series z + z^2 + z^4 + ... blows up at every dyadic root of unity, so no arc of the unit circle can be crossed.
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Proof Entire Functions of Polynomial Growth
An entire function with a bounded k-th derivative, or growth at most |z|^k, is a polynomial of degree at most k.