Physical Sciences and Mathematics Commons^™

A Weighted Version Of Erdős-Kac Theorem, Unique Subedi

Honors Theses

Let $\omega(n)$ denote the number of distinct prime factors of a natural number $n$. A celebrated result of Erd{\H o}s and Kac states that $\omega(n)$ as a Gaussian distribution. In this thesis, we establish a weighted version of Erd{\H o}s-Kac Theorem. Specifically, we show that the Gaussian limiting distribution is preserved, but shifted, when $\omega(n)$ is weighted by the $k-$fold divisor function $\tau_k(n)$. We establish this result by computing all positive integral moments of $\omega(n)$ weighted by $\tau_k(n)$.

We also provide a proof of the classical identity of $\zeta(2n)$ for $n \in \mathbb{N}$ using Dirichlet's kernel.

2-Adic Valuations Of Square Spiral Sequences, Minh Nguyen

Honors Theses

The study of p-adic valuations is connected to the problem of factorization of integers, an essential question in number theory and computer science. Given a nonzero integer n and prime number p, the p-adic valuation of n, which is commonly denoted as νp(n), is the greatest non-negative integer ν such that p ν | n. In this paper, we analyze the properties of the 2-adic valuations of some integer sequences constructed from Ulam square spirals. Most sequences considered were diagonal sequences of the form 4n 2 + bn + c from the Ulam spiral with center value of 1. Other …

An Introduction To Obstacle Problems, Calvin Reedy

Honors Theses

The obstacle problem can be used to predict the shape of an elastic membrane lying over an obstacle in a domain Ω. In this paper we introduce and motivate a mathematical formulation for this problem, and give an example to demonstrate the need to search for solutions in non-classical settings. We then introduce Sobolev spaces as the proper setting for solutions, and prove that unique solutions exist in W1,2(Ω).

Overdose Prevention Sites Placement Informed By Simulation, Jing Dong

Honors Theses

In Philadelphia, people are experiencing the greatest opioid crisis in a century. Plac- ing the Overdose Prevention Site (OPS) can alleviate this crisis. However, the journey to the successful launch of the first OPS in the USA is rough. It was first accused of having a collision with federal drug laws. While Safehouse won the lawsuit and the OPS was judged to be legal in 2020, other pressure rose afterward such as the against from the public and the COVID19, which delayed the plan to open the OPS. Without solid research on the effectiveness of OPS, we thought it is …

Computational Difficulty And Invariants Of The Snake Cube Puzzle, Adrian Negrea

Honors Theses

The snake cube is a popular puzzle that has been analyzed for its computational difficulty and shown to be NP-complete. Conceiving of the puzzle as a Hamiltonian path in an n x n x n graph, we offer a novel mathematical analysis by considering invariants of the puzzle. This allows us to determine necessary conditions for a particular snake cube to be solvable and eliminate a large class of possible puzzles as unsolvable. In particular, we establish upper and lower bounds on the possible number of straight components in solvable snake cube puzzles.

Dna Curve Classification With Unsupervised Learning, Ben Thomas

Honors Theses

No abstract provided.

Using Difference-In-Differences Analysis And The Kocyk Geometric Lag Model To Estimate Aspects Of Carbon Tax Effectiveness In Nordic Countries, Kyle Riley

Honors Theses

This paper generally looks at the connections between carbon taxes and carbon emission levels in Nordic countries over a period from the 1960s to the early 2010s. Most of the existing literature on this topic looks at and finds that carbon taxes do have a significant impact upon carbon emissions levels in some countries while not in others. In many countries which have this policy there is not a significant impact that can be seen and there is a discussion as to why this might be the case and what needs to be done to fix these potential issues to …

Counting Conjugacy Classes Of Elements Of Finite Order In Compact Exceptional Groups, Qidong He

Honors Theses

Given a compact exceptional group $G$ and $m,s\in\mathbb{N}$, let $N(G,m)$ be the number of conjugacy classes of elements of order $m$ in $G$, and $N(G,m,s)$ the number of such classes whose elements have $s$ distinct eigenvalues. In string theory, the problem of enumerating certain classes of vacua in the string landscape can be rephrased in terms of the study of these quantities. We develop unified combinatorial algorithms based on Burnside's Lemma that can be used to compute both quantities for each of the five compact exceptional groups. Thus, we provide a combinatorial, alternative method to that of Djoković and extend …

A Generalized Polar-Coordinate Integration Formula, Oscillatory Integral Techniques, And Applications To Convolution Powers Of Complex-Valued Functions On $\Mathbb{Z}^D$, Huan Q. Bui

Honors Theses

In this thesis, we consider a class of function on $\mathbb{R}^d$, called positive homogeneous functions, which interact well with certain continuous one-parameter groups of (generally anisotropic) dilations. Generalizing the Euclidean norm, positive homogeneous functions appear naturally in the study of convolution powers of complex-valued functions on $\mathbb{Z}^d$. As the spherical measure is a Radon measure on the unit sphere which is invariant under the symmetry group of the Euclidean norm, to each positive homogeneous function $P$, we construct a Radon measure $\sigma_P$ on $S=\{\eta \in \mathbb{R}^d:P(\eta)=1\}$ which is invariant under the symmetry group of $P$. With this measure, we prove …