Year
2026
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…
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…
In this paper we show that the chromatic function $P(G,t)$ is always a polynomial. We then ask a simple question: where does this polynomial equal zero? Tutte showed that two stretches of the real line, $(-\infty,0)$ and $(0,1)$, can never…
This paper primarily discusses the proof of the Mahler-Lech-Skolem through the uses of p-adic analysis in its connection to Strassman's Theorem and p-adic interpolation. As this theorem is so closely related to p-adic analysis, we make a b…
Combinatorial game theory and coding theory appear to be completely separate fields at first glance. However, they share a profound connection through structures known as lexicographic codes, or lexicodes. First introduced by Conway and Sl…
Hyperbolic geometry is a non-Euclidean geometry where geodesics play the role of straight lines. It is characterised by a constant negative curvature. The idea was conceived when mathematicians tried to understand the parallel postulate. T…
In statistics, signal processing, and machine learning, the information we want from a matrix is carried by its singular values and singular vectors, yet the matrix we observe is almost never the one we want: it has been corrupted by measu…
A tropical curve is a graph with specified edge lengths, some of which may be infinite. Various facts and attributes about algebraic curves have analogs for tropical curves. In this article, we focus on divisors and linear series, and prov…
This paper tries to create an intuition behind the proof of the fundamental theorem–Weierstrass approximation theorem–which will be shown using Bernstein polyno- mials and its generalization beyond polynomials–the Stone-Weierstrass theor…
An essay that presents a first-principles approach to eigenvalue dynamics of the wave equation applied in context of timpani tuning.
This expository paper explains tessellations of the hyperbolic plane from a differential-geometric viewpoint. We review the upper half-plane (H) and Poincaré disk (D) models and the Cayley transform linking them. We prove that orientation-…
We validate an inverse-drag estimation pipeline for finned spherical projectiles and, within this experimental regime, detect a robust non-monotonic effective-drag anomaly near L≈1.00. The pipeline pairs a custom fourth-order Runge–Kutta t…
In this paper, we begin with the concept of a projection in the familiar vector space R^n, and then extend it to an arbitrary inner product space. This foundational idea enables us to compute a line of best fit, approximate functions using…
In this paper, we explore matroids, which generalize properties of linear independence of vectors and cycles in graphs. We investigate properties and operations on matroids, then we look at how we can represent matroids using matrices.
In this article, we show that locally compact groups have invariant measures, also known as Haar measures. We also show that such measures are unique up to scaling.