Mathematics Commons^™

Uniformly Connected Graphs, Nasreen Almohanna

Dissertations

Perhaps the most fundamental property that a graph can possess is that of being connected. Two vertices u and v of a graph G are connected if G contains a u-v path. The graph G itself is connected if every two vertices of G are connected. The well-studied concept of connectivity provides a measure on how strongly connected a graph may be. There are many other degrees of connectedness for a graph. A Hamiltonian path in a graph G is a path containing every vertex of G. Among the best-known classes of highly connected graph are the Hamiltonian-connected graphs, …

Congruence Relations Mod 2 For (2 X 4^T + 1)-Colored Partitions, Nicholas Torello

Senior Theses

Let p_r(n) denote the difference between the number of r-colored partitions of n into an even number of distinct parts and into an odd number of distinct parts. Inspired by proofs involving modular forms of the Hirschhorn-Sellers Conjecture, we prove a similar congruence for p_r(n). Using the Jacobi Triple Product identity, we discover a much stricter congruence for p_3(n).

The Hyperreals: Do You Prefer Non-Standard Analysis Over Standard Analysis?, Chloe Munroe

Mathematics Undergraduate Theses

The hyperreal number system *ℝ forms an ordered field that contains ℝ as a subfield as well as infinitely many large and small numbers. A number is defined to be infinitely large if |ω| > n for all n = 1, 2, 3, ... and infinitely small if |ε| < 1/n for all n = 1, 2, 3... This number system is built out of the real number system analogous to Cantor’s construcion of ℝ out of ℚ. The new entities in *ℝ and the relationship between the reals and hyperreals provides an appealing alternate approach to real (standard) analysis …

The Name Tag Problem, Christian Carley

Mathematics Undergraduate Theses

A group of n people sit around a table, according to an assignment of name tags in which only one person is paired with the correct name tag. Curious to see if it will improve the number of correct pairings, everybody passes their name tag to the person on their left. Oddly, a new person, and only that person, receives the correct name tag. Indeed, every rotation provides the correct name tag to exactly one new person, until the nth rotation, whereby every person has received the correct name tag one at a time. Given any group of people, …

Accurate Inference For Repeated Measures In High Dimensions, Xiaoli Kong, Solomon W. Harrar

Mathematics and Statistics: Faculty Publications and Other Works

This paper proposes inferential methods for high-dimensional repeated measures in factorial designs. High-dimensional refers to the situation where the dimension is growing with sample size such that either one could be larger than the other. The most important contribution relates to high-accuracy of the methods in the sense that p-values, for example, are accurate up to the second-order. Second-order accuracy in sample size as well as dimension is achieved by obtaining asymptotic expansion of the distribution of the test statistics, and estimation of the parameters of the approximate distribution with second-order consistency. The methods are presented in a unified and …

Confidence Intervals For The Area Under The Receiver Operating Characteristic Curve In The Presence Of Ignorable Missing Data, Hunyong Cho, Gregory J. Matthews, Ofer Harel

Mathematics and Statistics: Faculty Publications and Other Works

Receiver operating characteristic curves are widely used as a measure of accuracy of diagnostic tests and can be summarised using the area under the receiver operating characteristic curve (AUC). Often, it is useful to construct a confidence interval for the AUC; however, because there are a number of different proposed methods to measure variance of the AUC, there are thus many different resulting methods for constructing these intervals. In this article, we compare different methods of constructing Wald‐type confidence interval in the presence of missing data where the missingness mechanism is ignorable. We find that constructing confidence intervals using multiple …

Deciphering The Transport Of Elastic Filaments By Antagonistic Motor Proteins, Stephanie Portet, Cecil Leduc, Sandrine Etienne-Manneville, J. C. Dallon

Faculty Publications

Intermediate filaments are long elastic fibres that are transported by microtubule-associated motor proteins kinesin and dynein inside the cell. How elastic filaments are efficiently transported by antagonistic motors is not well understood and difficult to measure with current experimental techniques. Adapting the tug-of-war paradigm for vesicle-like cargos, we develop a mathematical model to describe the motion of an elastic filament punctually bound to antagonistic motors. As observed in cells, up to 3 modes of transport are obtained; dynein-driven retrograde, kinesin-driven anterograde fast motions and a slow motion. Motor properties and initial conditions that depend on intracellular context, regulate the transport …

Extension Of Soft Set To Hypersoft Set, And Then To Plithogenic Hypersoft Set, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

In this paper, we generalize the soft set tothe hypersoft set by transforming the function F into a multi-attribute function. Then we introduce the hybrids of Crisp, Fuzzy, Intuitionistic Fuzzy, Neutrosophic, and Plithogenic Hypersoft Set.

Sequential Differences In Nabla Fractional Calculus, Ariel Setniker

Department of Mathematics: Dissertations, Theses, and Student Research

We study the composition of nabla fractional differences of unequal orders, known as "sequential" nabla fractional differences. The sequential differences we examine possess different bases — specifically, we establish the outer operator as having a base larger than the inner operator by at least an integer factor of 1. Further, we consider two cases of orders: first the case when the outer difference has a larger power, and second when the inner difference has a larger power.

We develop rules for sequential nabla fractional differences and present connections between the sign of a sequential difference of a function and the …

Variations In Ramsey Theory, Drake Olejniczak

Dissertations

The Ramsey number R(F,H) of two graphs F and H is the smallest positive integer n for which every red-blue coloring of the (edges of a) complete graph of order n results in a graph isomorphic to F all of whose edges are colored red (a red F) or a blue H. Beineke and Schwenk extended this concept to a bipartite version of Ramsey numbers, namely the bipartite Ramsey number BR(F,H) of two bipartite graphs F and H is the smallest positive integer rsuch that every red-blue coloring of the r-regular complete bipartite graph results in either …

Integrating Mathematics And Educational Robotics: Simple Motion Planning, Ronald I. Greenberg, George K. Thiruvathukal, Sara T. Greenberg

Computer Science: Faculty Publications and Other Works

This paper shows how students can be guided to integrate elementary mathematical analyses with motion planning for typical educational robots. Rather than using calculus as in comprehensive works on motion planning, we show students can achieve interesting results using just simple linear regression tools and trigonometric analyses. Experiments with one robotics platform show that use of these tools can lead to passable navigation through dead reckoning even if students have limited experience with use of sensors, programming, and mathematics.

The Number Of Fixed Points Of And-Or Networks With Chain Topology, Lauren Geiser

Honors Theses

Boolean networks are sets of Boolean functions, which are functions that contain Boolean variables and the logical operators AND, OR, and NOT. In the simple case, the variables can be in one of two states—either 1 or 0, which can be interpreted in different ways such as ON or OFF, or TRUE or FALSE, depending on the application. Arranging model systems into Boolean functions, we can study steady states of these networks. This refers to the overall state of the dynamical system given an initial condition and another theoretical condition such as a subsequent point in time. Boolean networks have …

Topology Of Fractals, Amelia Pompilio

Honors Theses

No abstract provided.

Cryptography: Mathematical Advancements On Cyber Security, Kristin Bower

Senior Honors Theses

The origin of cryptography, the study of encoding and decoding messages, dates back to ancient times around 1900 BC. The ancient Egyptians enlisted the use of basic encryption techniques to conceal personal information. Eventually, the realm of cryptography grew to include the concealment of more important information, and cryptography quickly became the backbone of cyber security. Many companies today use encryption to protect online data, and the government even uses encryption to conceal confidential information. Mathematics played a huge role in advancing the methods of cryptography. By looking at the math behind the most basic methods to the newest methods …

Dynamical Entropy Of Quantum Random Walks, Duncan Wright

Theses and Dissertations

In this manuscript, we study discrete-time dynamics of systems that arise in physics and information theory, and the measure of disorder in these systems known as dy- namical entropy. The study of dynamics in classical systems is done from two distinct viewpoints: random walks and dynamical systems. Random walks are probabilistic in nature and are described by stochastic processes. On the other hand, dynami- cal systems are described algebraically and deterministic in nature. The measure of disorder from either viewpoint is known as dynamical entropy.

Entropy is an essential notion in physics and information theory. Motivated by the study of …

Determinants Of Incidence And Hessian Matrices Arising From The Vector Space Lattice, Saeed Nasseh, Alexandra Seceleanu, Junzo Watanabe

Department of Mathematics: Faculty Publications

Let V = ⁿi= _o V^I bethe lattice of subspaces of the n-dimensional vector space over the finite field Fq, and let A be the graded Gorenstein algebra defined over Q which has V as a Q basis. Let F be the Macaulay dual generator for A. We explicitly compute the Hessian determinant j 2F= Xi Xj j, evaluated at the point X1 = X2 = ... = XN = 1, and relate it to the determinant of the incidence matrix between V1 and Vn-1. Our exploration is motivated by the fact that both of these matrices naturally …

On The Characteristic Polynomial Of A Hypergraph, Gregory J. Clark

Theses and Dissertations

We consider the computation of the adjacency characteristic polynomial of a uniform hypergraph. Computing the aforementioned polynomial is intractable, in general; however, we present two mechanisms for computing partial information about the spectrum of a hypergraph as well as a methodology (and in particular, an algo- rithm) for combining this information to determine complete information about said spectrum. The first mechanism is a generalization of the Harary-Sachs Theorem for hypergraphs which yields an explicit formula for each coefficient of the characteristic polynomial of a hypergraph as a weighted sum over a special family of its subgraphs. The second is a …

Successful Pressing Sequences In Simple Pseudo-Graphs, Hays Wimsatt Whitlatch

Theses and Dissertations

Motivated by the study of genomes evolving by reversals, the primary topic of this thesis is “successful pressing sequences” in simple pseudo-graphs. Pressing sequences where first introduced by Hannenhali and Pevzner in 1999 where they showed that sorting signed permutation problem can be solved in polynomial time, therefore demonstrating that the length of a most parsimonious solution to the genome in- version only rearrangement problem can be determined efficiently.

A signed permutation is an integer permutation where each entry is given a sign: plus or minus. A reversal in a signed permutation is the operation of reversing a subword and …

On The Generators Of Quantum Dynamical Semigroups, Alexander Wiedemann

Theses and Dissertations

In recent years, digraph induced generators of quantum dynamical semigroups have been introduced and studied, particularly in the context of unique relaxation and invariance. We define the class of pair block diagonal generators, which allows for additional interaction coefficients but preserves the main structural properties. Namely, when the basis of the underlying Hilbert space is given by the eigenbasis of the Hamiltonian (for example the generic semigroups), then the action of the semigroup leaves invariant the diagonal and off-diagonal matrix spaces. In this case, we explicitly compute all invariant states of the semigroup.

In order to define this class we …

A New Characterization Of V-Posets, Peter Gartland

Theses and Dissertations

In 2016, Hasebe and Tsujie gave a recursive characterization of the set of induced N -free and bowtie-free posets; Misanantenaina and Wagner studied these orders fur- ther, naming them “V-posets”. Here we offer a new characterization of V-posets by introducing a property we refer to as autonomy. A poset P is said to be autonomous if there exists a directed acyclic graph D (with adjacency matrix U ) whose transitive closure is P, with the property that any total ordering of the vertices of D so that Gaussian elimination of UT U proceeds without row swaps is a …