Year
2026
In this paper, we use multiple techniques to find the complete set of solutions to Pell and negative Pell equations. We prove the existence of a nontrivial solution to the former using Dirichlet's Approximation Theorem and then find all so…
Sparse linear algebra is the study of linear systems Ax = b in which the matrix A contains overwhelmingly many zero entries. Such systems arise throughout applied mathematics and engineering from the discretization of partial differential…
The representations of the Symmetric Groups S_n carry a very rich combinatorial structure. In particular, their representations have connections to seemingly unrelated problems, including to symmetric polynomials and the theory of Young Ta…
This paper builds up and proves the Poincaré-Hopf Theorem for compact smooth manifolds. Beginning with the introduction of key concepts in differential topology, including smooth manifolds, coordinate charts, tangent spaces, and mappings, …
The symmetric groups are fundamental objects in abstract algebra, and for nearly every value of n, their automorphism groups are completely inner. The unique exception is the symmetric group S_6, which possesses a nontrivial outer autom…
This paper gives an expository introduction to arithmetic progressions in Ramsey theory and additive combinatorics. We begin with the language of colorings, using small examples to illustrate how simple coloring rules can force surprisingl…
In this paper, we will introduce Fourier Series through the Orthogonality Relations from which we consider Parseval's Theorem. Then, we wish to extend the concept of Fourier Series of aperiodic functions, specifically showing the Fourier T…
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 importan…
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 throug…
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 polyt…
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 scenar…
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.