Physical Sciences and Mathematics Commons^™

On The Construction And Mathematical Analysis Of The Wavelet Transform And Its Matricial Properties, Diego Sejas Viscarra

Rose-Hulman Undergraduate Mathematics Journal

We study the properties of computational methods for the Wavelet Transform and its Inverse from the point of view of Linear Algebra. We present a characterization of such methods as matrix products, proving in particular that each iteration corresponds to the multiplication of an adequate unitary matrix. From that point we prove that some important properties of the Continuous Wavelet Transform, such as linearity, distributivity over matrix multiplication, isometry, etc., are inherited by these discrete methods.

This work is divided into four sections. The first section corresponds to the classical theoretical foundation of harmonic analysis with wavelets; it is used …

Dna Self-Assembly Design For Gear Graphs, Chiara Mattamira

Rose-Hulman Undergraduate Mathematics Journal

Application of graph theory to the well-known complementary properties of DNA strands has resulted in new insights about more efficient ways to form DNA nanostructures, which have been discovered as useful tools for drug delivery, biomolecular computing, and biosensors. The key concept underlying DNA nanotechnology is the formation of complete DNA complexes out of a given collection of branched junction molecules. These molecules can be modeled in the abstract as portions of graphs made up of vertices and half-edges, where complete edges are representations of double-stranded DNA pieces that have joined together. For efficiency, one aim is to minimize the …

Generalised Fibonacci Sequences Under Modular Arithmetic, Connor Riddlesden

Rose-Hulman Undergraduate Mathematics Journal

In this paper, we find patterns and count the number of distinct generalised Fibonacci sequences under modular arithmetic. We will start with the repetition of the normal Fibonacci sequence modulo an integer, m, where m is greater than or equal to two and make connections to its dependency on the prime factorisation of m. We will then extend the complexity of the problem into generalised Fibonacci sequences with different starting values. Finally we will present some interesting observations that are still open problems.

The Name Tag Problem, Christian Carley

Rose-Hulman Undergraduate Mathematics Journal

The Name Tag Problem is a thought experiment that, when formalized, serves as an introduction to the concept of an orthomorphism of $\Zn$. Orthomorphisms are a type of group permutation and their graphs are used to construct mutually orthogonal Latin squares, affine planes and other objects. This paper walks through the formalization of the Name Tag Problem and its linear solutions, which center around modular arithmetic. The characterization of which linear mappings give rise to these solutions developed in this paper can be used to calculate the exact number of linear orthomorphisms for any additive group Z/nZ, which is demonstrated …

Configuration Spaces For The Working Undergraduate, Lucas Williams

Rose-Hulman Undergraduate Mathematics Journal

Configuration spaces form a rich class of topological objects which are not usually presented to an undergraduate audience. Our aim is to present configuration spaces in a manner accessible to the advanced undergraduate. We begin with a slight introduction to the topic before giving necessary background on algebraic topology. We then discuss configuration spaces of the euclidean plane and the braid groups they give rise to. Lastly, we discuss configuration spaces of graphs and the various techniques which have been developed to pursue their study.

On The Equitable Total (𝑘+1)-Coloring Of 𝑘-Regular Graphs, Bryson Stemock

Rose-Hulman Undergraduate Mathematics Journal

A graph is considered to be totally colored when one color is assigned to each vertex and to each edge so that no adjacent or incident vertices or edges bear the same color. The \textit{total chromatic number} of a graph is the least number of colors required to totally color a graph. This paper focuses on $k$-regular graphs, whose symmetry and regularity allow for a closer look at general total coloring strategies. Such graphs include the previously defined M\"obius ladder, which has a total chromatic number of 5, as well as the newly defined bird's nest, which is shown to …

Colorings And Sudoku Puzzles, Katelyn D. May

Rose-Hulman Undergraduate Mathematics Journal

Map colorings refer to assigning colors to different regions of a map. In particular, a typical application is to assign colors so that no two adjacent regions are the same color. Map colorings are easily converted to graph coloring problems: regions correspond to vertices and edges between two vertices exist for adjacent regions. We extend these notions to Shidoku, 4x4 Sudoku puzzles, and standard 9x9 Sudoku puzzles by demanding unique entries in rows, columns, and regions. Motivated by our study of ring and field theory, we expand upon the standard division algorithm to study Gr\"obner bases in multivariate polynomial rings. …

Investigating First Returns: The Effect Of Multicolored Vectors, Shakuan Frankson, Myka Terry

Rose-Hulman Undergraduate Mathematics Journal

By definition, a first return is the immediate moment that a path, using vectors in the Cartesian plane, touches the x-axis after leaving it previously from a given point; the initial point is often the origin. In this case, using certain diagonal and horizontal vectors while restricting the movements to the first quadrant will cause almost every first return to end at the point (2n,0), where 2n counts the equal number of up and down steps in a path. The exception will be explained further in the sections below. Using the first returns of Catalan, Schröder, and Motzkin numbers, which …

New Theorems For The Digraphs Of Commutative Rings, Morgan Bounds

Rose-Hulman Undergraduate Mathematics Journal

The digraphs of commutative rings under modular arithmetic reveal intriguing cycle patterns, many of which have yet to be explained. To help illuminate these patterns, we establish a set of new theorems. Rings with relatively prime moduli a and b are used to predict cycles in the digraph of the ring with modulus ab. Rings that use Pythagorean primes as their modulus are shown to always have a cycle in common. Rings with perfect square moduli have cycles that relate to their square root.

On The Enumeration Of Shapes, May Cai, Nicholas Liao

Rose-Hulman Undergraduate Mathematics Journal

We define a shape as a union of finitely many line segments. Given an arrangement of lines on a plane, we count the number of shapes in the arrangement by examining the symmetries of the arrangement and applying Burnside's lemma. We further establish a generating function for the number of distinct line segments on a line with k distinguished points. We list all affine line arrangements of four and five line segments, together with the corresponding number of shapes on them.

Iterated Line Graphs On Bi-Regular Graphs And Trees, Brenden Balch

Rose-Hulman Undergraduate Mathematics Journal

In 1965, van Rooij and Wilf considered sequences of line graphs, in which they grouped sequences of line graphs into four categories. We’ll add to their research by presenting results on sequences of line graphs for star graphs and bi-regular graphs. We will then investigate slight variations of star graphs.

A Connection Between Quadratic Rational Maps And Linear Fractional Maps, Laura Schlesinger, Anna Marek, Ella White, Danqi Yin

Rose-Hulman Undergraduate Mathematics Journal

This research project is an investigation into quadratic rational maps, $\vp$, of one complex variable that map the unit disk to itself. Previous research \cite{brittney} shows that for each $\vp$, a corresponding linear fractional map $\zeta$ can be found using the coefficients of $\vp$, and this $\zeta$ can be used to characterize functions in the kernel of the adjoint of the composition operator with symbol $\vp$, defined on a space of analytic functions. In this paper, we show sufficient conditions to ensure that certain cases of $\vp$ map the unit disk to itself and find all the forms of $\zeta$. …

𝑘-Plane Constant Curvature Conditions, Maxine E. Calle

Rose-Hulman Undergraduate Mathematics Journal

This research generalizes the two invariants known as constant sectional curvature (csc) and constant vector curvature (cvc). We use k-plane scalar curvature to investigate the higher-dimensional analogues of these curvature conditions in Riemannian spaces of arbitrary finite dimension. Many of our results coincide with the known features of the classical k=2 case. We show that a space with constant k-plane scalar curvature has a uniquely determined tensor and that a tensor can be recovered from its k-plane scalar curvature measurements. Through two example spaces with canonical tensors, we demonstrate a method for determining constant k-plane …

Isoperimetric Problems On The Line With Density |𝑥|ᵖ, Juiyu Huang, Xinkai Qian, Yiheng Pan, Mulei Xu, Lu Yang, Junfei Zhou

Rose-Hulman Undergraduate Mathematics Journal

On the line with density |x|^p, we prove that the best single bubble is an interval with endpoint at the origin and that the best double bubble is two adjacent intervals that meet at the origin.

The Isoperimetric Inequality: Proofs By Convex And Differential Geometry, Penelope Gehring

Rose-Hulman Undergraduate Mathematics Journal

The Isoperimetric Inequality has many different proofs using methods from diverse mathematical fields. In the paper, two methods to prove this inequality will be shown and compared. First the 2-dimensional case will be proven by tools of elementary differential geometry and Fourier analysis. Afterwards the theory of convex geometry will briefly be introduced and will be used to prove the Brunn--Minkowski-Inequality. Using this inequality, the Isoperimetric Inquality in n dimensions will be shown.

On Consecutive Triples Of Powerful Numbers, Edward Beckon

Rose-Hulman Undergraduate Mathematics Journal

A powerful number is a positive integer such that every prime that appears in its prime factorization appears there at least twice. Erdős, Mollin and Walsh conjectured that three consecutive powerful numbers do not exist. This paper shows that if they do exist, the smallest of the three numbers must have remainder 7, 27, or 35 when divided by 36.

Consecutive Prime And Highly Total Prime Labeling In Graphs, Robert Scholle

Rose-Hulman Undergraduate Mathematics Journal

This paper examines the graph-theoretical concepts of consecutive prime labeling and highly total prime labeling. These are variations on prime labeling, introduced by Tout, Dabboucy, and Howalla in 1982. Consecutive prime labeling is defined here for the first time. Consecutive prime labeling requires that the labels of vertices in a graph be relatively prime to the labels of all adjacent vertices as well as all incident edges. We show that all paths, cycles, stars, and complete graphs have a consecutive prime labeling and conjecture that all simple connected graphs have a consecutive prime labeling.

This paper also expands on work …

Combinatorial Identities On Multinomial Coefficients And Graph Theory, Seungho Lee

Rose-Hulman Undergraduate Mathematics Journal

We study combinatorial identities on multinomial coefficients. In particular, we present several new ways to count the connected labeled graphs using multinomial coefficients.