On the Singular Pebbling Number of a Graph

In this paper, we define a new parameter of a connected graph as a spin-off of the pebbling number (which is the smallest t such that every supply of t pebbles can satisfy every demand of one pebble). This new parameter is the singular pebbling number, the smallest t such that a player can be given any configuration of at least t pebbles and any target vertex and can successfully move pebbles so that exactly one pebble ends on the target vertex. We also prove that the singular pebbling number of any graph on 3 or more vertices is equal to its pebbling number, that the singular pebbling number of the disconnected graph on two vertices is equal to its pebbling number, and we find the singular pebbling numbers of the two remaining graphs, K1 and K2, which are not equal to their pebbling numbers

Further Generalizations of Happy Numbers

A positive integer n is defined to be happy if iteration of the function taking the sum of the squares of the digits of n eventually reaches 1. In this paper we generalize the concept of happy numbers in several ways. First we confirm known results of Grundman and Teeple and establish further results extending the known structure of happy numbers to higher powers. Then we construct a similar function expanding the definition of happy numbers to negative integers. Working with this function, we prove a range of results paralleling those already proven for traditional and generalized happy numbers. Finally, we consider a variety of special cases, in which the existence of certain fixed points and cycles of infinite families of generalized happy functions can be proven

The Existence of Solutions to a System of Nonhomogeneous Difference Equations

This article will demonstrate a process using Fixed Point Theory to determine the existence of multiple positive solutions for a type of system of nonhomogeneous even ordered boundary value problems on a discrete domain. We first reconstruct the problem by transforming the system so that it satisfies homogeneous boundary conditions. We then create a cone and an operator sufficient to apply the Guo-KrasnoselâA˘Zskii Fixed Point Theorem. The majority of the work involves developing the constraints ´ needed to utilized this fixed point theorem. The theorem is then applied three times, guaranteeing the existence of at least three distinct solutions. Thus, solutions to this class of boundary value problems exist and are not unique

A Note On The Involutive Concordance Invariants For Certain (1,1)-Knots

A knot K is a smooth embedding of the circle into the three-dimensional sphere; two knots are said to be concordant if they form the boundary of an annulus properly embedded into the product of the three-sphere with an interval. Heegaard Floer knot homology is an invariant of knots introduced by P. Ozsváth and Z. Szabó in the early 2000\u27s which associates to a knot a filtered chain complex CFK(K), which improves on classical invariants of the knot. Involutive Heegaard Floer homology is a variant theory introduced in 2015 by K. Hendricks and C. Manolescu which additionally considers a chain map iota on CFK(K) induced by a conjugation operation, and extracts from this data two new numerical invariants of knot concordance. These new invariants are especially interesting because, unlike many other concordance invariants from Heegaard Floer homology, they do not necessarily vanish on knots of finite order in the group of concordance classes of knots. The chain map iota is in general difficult to compute, and computations have been carried out for relatively few knots. We give a complete computation of iota for 10 and 11-crossing knots satisfying a certain simplicity condition, called the (1,1)-knots

k-Distinct Lattice Paths

Lattice paths can be used to model scheduling and routing problems, and, therefore, identifying maximum sets of k-distinct paths is of general interest. We extend the work previously done by Gillman et. al. to determine the order of a maximum set of k-distinct lattice paths. In particular, we disprove a conjecture by Gillman that a greedy algorithm gives this maximum order and also refine an upper bound given by Brewer et. al. We illustrate that brute force is an inefficient method to determine the maximum order, as it has time complexity O(nk)

The Chromatic Index of Ring Graphs

The goal of graph edge coloring is to color a graph G with as few colors as possible such that each edge receives a color and that adjacent edges, that is, different edges incident to a common vertex, receive different colors. The chromatic index, denoted χ′(G), is the minimum number of colors required for such a coloring to be possible. There are two important lower bounds for χ′(G) on every graph: maximum degree, denoted ∆(G), and density, denoted ω(G). Combining these two lower bounds, we know that every graph’s chromatic index must be at least ∆(G) or ω(G), whichever is greater. In this paper, we prove that the chromatic index of every ring graph is exactly equal to this lower bound

Optical Metrology and Beam Shaping using Active Lensing and Engineered Diffusers

This thesis encompasses three chapters that delve into innovative studies within optical experiments. The overarching motivation behind this research is to enhance the efficiency, and versatility of measurement techniques used in the field. Furthermore, by exploring novel technologies and methodologies, these studies aim to overcome existing limitations and open new avenues for optical experimentation. The first chapter addresses the challenge of accurately measuring the radius of curvature of a sample without physically moving the lens position. Conventional methods using single lenses are prone to errors caused by mechanical drift or instability. To mitigate these issues, the study introduces Tunable Focal Lenses (TFLs) as a replacement for fixed focal length lenses. By employing TFLs in Twyman-Green and Fizeau interferometry, the research demonstrates a more reliable and expeditious approach. The experimental results highlight the potential of TFLs to eliminate errors associated with physical lens movement and present a promising alternative to conventional lens configurations. In the second chapter, the focus shifts to improving diffusing systems utilized in optical experiments. The motivation behind this study lies in the need for efficient and effective replacements for pinhole-based diffusers. Engineered Diffusers (EDs), with their advanced properties, emerge as a potential solution. Specifically, the research delves into utilizing the Light Shaping Diffuser (LSD) due to its high light transmission rate. The experiment explores the spatial characteristics of a collimated beam diffused by LSD and investigates methods to mitigate speckle patterns that often arise. To this end, the research introduces the Speckle Reducer (SR), which combines liquid crystal technology and diffractive optics. Through a comprehensive analysis utilizing techniques such as CCD imaging, Shack-Hartmann wavefront measurement, and knife-edge measurement, the study provides valuable insights into beam divergence calculation, beam profile analysis, and wavefront curvature measurement. By showcasing the potential of EDs, particularly LSDs, as viable alternatives to pinhole-based diffusing systems, this research paves the way for enhanced optical experiments. The final chapter delves into the experimental exploration of LSDs and SRs for interferometry measurements. The primary motivation is to evaluate the effectiveness of this combination as an adaptive optical measurement tool. A Michelson interferometer is employed to evaluate, wherein a beam shaped by LSD and enhanced by SR is split using a beam splitter. The divided parts are directed towards a reference mirror and an adjustable mirror, respectively, before being recombined at the beam splitter. The resulting fringe pattern is captured by a CCD camera, enabling detailed analysis. Moreover, the study investigates the measurement of the refractive index of a thick glass placed between the beam splitter and mirror by utilizing the enhanced beam\u27s bull\u27s eye fringe pattern generated by the SR. The experiment aims to observe changes in fringe patterns as the glass is tilted at various angles, offering valuable insights into the potential applications of LSDs and SRs in interferometry measurements. Through these three interconnected chapters, this research project presents a comprehensive exploration of cutting-edge techniques and technologies in optical experiments. By addressing key challenges and proposing innovative solutions, this work contributes to advancing measurement methodologies, offering new perspectives and possibilities for the optical research community

Utilizing graph thickness heuristics on the Earth-moon Problem

This paper utilizes heuristic algorithms for determining graph thickness in order to attempt to find a 10-chromatic thickness-2 graph. Doing so would eliminate 9 colors as a potential solution to the Earth-moon Problem. An empirical analysis of the algorithms made by the author are provided. Additionally, the paper lists various graphs that may or nearly have a thickness of 2, which may be solutions if one can find two planar subgraphs that partition all of the graph’s edges

Divisibility Probabilities for Products of Randomly Chosen Integers

We find a formula for the probability that the product of n positive integers, chosen at random, is divisible by some integer d. We do this via an inductive application of the Chinese Remainder Theorem, generating functions, and several other combinatorial arguments. Additionally, we apply this formula to find a unique, but slow, probabilistic primality test