What is Expositio?
A public repository for expository academic papers, with an integrated scholarly record.
Expositio is a place to read and cite expository work. Papers are published as stable,
citable records, while remaining connected to later versions, editorial decisions,
and related context.
Some papers are early drafts; others have been peer reviewed or formally published.
Each paper remains readable on its own, while preserving the history behind it.
The result is both a reading room and a record: a place to browse by field, read
papers in stable form, and follow their development over time.
v1
Firstly, we will introduce some preliminary theory about Newton Polygons while presenting some comments for their uses. Our main focus is analyzing the Newton Polgons of general functions, specifically power series, of which a few important lemmas are shown. Finally, we collate …
Mathematics / Number theory
v1
This paper introduces Minimal Surfaces through the Weingarten Map and the 1st and 2nd Fundamental forms. Then, we consider two examples of a minimal surface, the catenoid and helicoid, verifying that both are indeed minimal surfaces through calculation. Finally, we end with two …
Mathematics / Differential geometry
v1
We investigate the geometric and combinatorial properties of polytope
triangulations, focusing on their connections to Ehrhart positivity.
The Ehrhart polynomial L(P,t) encodes the number of integer points in dilated
copies of a polytope, and establishing the positivity of …
Mathematics / Convex and discrete geometry
v1
This paper describes the mathematical technique of calculus of variations starting with the derivation of key equations such as the Euler-Lagrange equation and the Beltrami Identity. These equations are then applied in the following scenarios: geodesic on a flat plane, solving t…
Mathematics / Calculus of variations and optimal control; optimization
v1
The Fundamental Theorem of Calculus, Green's theorem, the classical Stokes
theorem, and the divergence theorem each relate differentiation on a region
to integration on its boundary. This paper develops the language of smooth
manifolds and differential forms in which these r…
Mathematics / Global analysis, analysis on manifolds
Mathematics / Algebraic geometry
Mathematics / Differential geometry
Mathematics / Linear and multilinear algebra; matrix theory
v1
Sidorenko's conjecture predicts that the homomorphism density of a bipartite graph H in a host graph G satisfies t_H(G) ≥ t_{K_2}(G)^|E(H)|. We introduce the conjecture through homomorphism densities, prove the finite-host inequality for P_3, C_4, and the complete bipartite grap…
Mathematics / Combinatorics
