Pascal's Triangle, Pascal's Pyramid, And The Trinomial Triangle, 2019 California State University, San Bernardino

#### Pascal's Triangle, Pascal's Pyramid, And The Trinomial Triangle, Antonio Saucedo Jr.

*Electronic Theses, Projects, and Dissertations*

Many properties have been found hidden in Pascal's triangle. In this paper, we will present several known properties in Pascal's triangle as well as the properties that lift to different extensions of the triangle, namely Pascal's pyramid and the trinomial triangle. We will tailor our interest towards Fermat numbers and the hockey stick property. We will also show the importance of the hockey stick properties by using them to prove a property in the trinomial triangle.

The Martingale Approach To Financial Mathematics, 2019 California Polytechnic State University, San Luis Obispo

#### The Martingale Approach To Financial Mathematics, Jordan M. Rowley

*Master's Theses and Project Reports*

In this thesis, we will develop the fundamental properties of financial mathematics, with a focus on establishing meaningful connections between martingale theory, stochastic calculus, and measure-theoretic probability. We first consider a simple binomial model in discrete time, and assume the impossibility of earning a riskless profit, known as arbitrage. Under this no-arbitrage assumption alone, we stumble upon a strange new probability measure *Q*, according to which every risky asset is expected to grow as though it were a bond. As it turns out, this measure *Q* also gives the arbitrage-free pricing formula for every asset on our market. In considering ...

Pairwise Completely Positive Matrices And Conjugate Local Diagonal Unitary Invariant Quantum States, 2019 Mount Allison University

#### Pairwise Completely Positive Matrices And Conjugate Local Diagonal Unitary Invariant Quantum States, Nathaniel Johnston, Olivia Maclean

*Electronic Journal of Linear Algebra*

A generalization of the set of completely positive matrices called pairwise completely positive (PCP) matrices is introduced. These are pairs of matrices that share a joint decomposition so that one of them is necessarily positive semidefinite while the other one is necessarily entrywise non-negative. Basic properties of these matrix pairs are explored and several testable necessary and sufficient conditions are developed to help determine whether or not a pair is PCP. A connection with quantum entanglement is established by showing that determining whether or not a pair of matrices is pairwise completely positive is equivalent to determining whether or not ...

Analyzing The Structural Properties Of Pulmonary Arterial Networks, 2019 North Carolina State University at Raleigh

#### Analyzing The Structural Properties Of Pulmonary Arterial Networks, Megan J. Chambers

*Biology and Medicine Through Mathematics Conference*

No abstract provided.

Optimal Spraying Strategies For Controlling Re-Infestation By Chagas Disease Vectors, 2019 Virginia Commonwealth University

#### Optimal Spraying Strategies For Controlling Re-Infestation By Chagas Disease Vectors, Bismark Oduro, Winfried Just, Mario Grijalva

*Biology and Medicine Through Mathematics Conference*

No abstract provided.

Stokes, Gauss, And Bayes Walk Into A Bar..., 2019 University of Colorado, Boulder

#### Stokes, Gauss, And Bayes Walk Into A Bar..., Eric P. Kightley

*Applied Mathematics Graduate Theses & Dissertations*

This thesis consists of three distinct projects. The first is a study of microbial aggregate fragmentation, in which we develop a dynamical model of aggregate deformation and breakage and use it to obtain a post-fragmentation density function. The second and third projects deal with dimensionality reduction in machine learning problems. In the second project, we derive a one-pass sparsified Gaussian mixture model to perform clustering analysis on high-dimensional streaming data. The model estimates parameters in dense space while storing and performing computations in a compressed space. In the final project, we build an expert system classifier with a Bayesian network ...

Characterizing The Tails Of Degree Distributions In Real-World Networks, 2019 University of Colorado, Boulder

#### Characterizing The Tails Of Degree Distributions In Real-World Networks, Anna Broido

*Applied Mathematics Graduate Theses & Dissertations*

This is a thesis about how to characterize the statistical structure of the tails of degree distributions of real-world networks. The primary contribution is a statistical test of the prevalence of scale-free structure in real-world networks. A central claim in modern network science is that real-world networks are typically "scale free," meaning that the fraction of nodes with degree k follows a power law, decaying like k^{-a}, often with 2 < a< 3. However, empirical evidence for this belief derives from a relatively small number of real-world networks. In the first section, we test the universality of scale-free structure by applying state-of-the-art statistical tools to a large corpus of nearly 1000 network data sets drawn from social, biological, technological, and informational sources. We fit the power-law model to each degree distribution, test its statistical plausibility, and compare it via a likelihood ratio test to alternative, non-scale-free models, e.g., the log-normal. Across domains, we find that scale-free networks are rare, with only 4% exhibiting the strongest-possible evidence of scale-free structure and 52% exhibiting the weakest-possible evidence. Furthermore, evidence of scale-free structure is not uniformly distributed across sources: social networks are at best weakly scale free, while a handful of technological and biological networks can be called strongly scale free. These results undermine the universality of scale-free networks and reveal that real-world networks exhibit a rich structural diversity that will likely require new ideas and mechanisms to explain. A core methodological component of addressing the ubiquity of scale-free structure in real-world networks is an ability to fit a power law to the degree distribution. In the second section, we numerically evaluate and compare, using both synthetic data with known structure and real-world data with unknown structure, two statistically principled methods for estimating the tail parameters for power-law distributions, showing that in practice, a method based on extreme value theory and a sophisticated bootstrap and the more commonly used method based an empirical minimization approach exhibit similar accuracy.

Proof Of A Conjecture Of Graham And Lovasz Concerning Unimodality Of Coefficients Of The Distance Characteristic Polynomial Of A Tree, 2019 Iowa State University

#### Proof Of A Conjecture Of Graham And Lovasz Concerning Unimodality Of Coefficients Of The Distance Characteristic Polynomial Of A Tree, Ghodratollah Aalipour, Aida Abiad, Zhanar Berikkyzy, Leslie Hogben, Franklin H.J. Kenter, Jephian C.-H. Lin, Michael Tait

*Leslie Hogben*

The conjecture of Graham and Lov ́asz that the (normalized) coefficients of the distance characteristic polynomial of a tree are unimodal is proved; it is also shown that the (normalized) coefficients are log-concave. Upper and lower bounds on the location of the peak are established.

Combinatorial Optimization: Introductory Problems And Methods, 2019 University of Connecticut

#### Combinatorial Optimization: Introductory Problems And Methods, Erin Brownell

*Honors Scholar Theses*

This paper will cover some topics of combinatorial optimization, the study of ﬁnding the best possible arrangement of a set of discrete objects. These topics include the shortest path problem and network ﬂows, which can be extended to solve more complex problems. We will also brieﬂy cover some basics of graph theory and solving linear programming problems to give context to the reader.

Math Modeling Contest: Optimized Plan To Leave The Louvre, 2019 Carroll College

#### Math Modeling Contest: Optimized Plan To Leave The Louvre, Shirley Davidson, Terry Cox, Sabrina Crooks

*Carroll College Student Undergraduate Research Festival*

The distances between every section of the Louvre, exiting speeds dependent upon an individual's surroundings, time of day, total people, distribution of people, and stair dimensions are all factors utilized to establish a model for evacuating the Louvre. Our goal was to create an emergency protocol procedure that minimized the amount of time required to safely remove all guests from the Louvre. Using Dijkstra’s Algorithm to find the shortest path based on average distance between connecting nodes, three different piecewise difference equations were created to model the flow of guests out of the Louvre. With an average of ...

Vector Spaces Of Generalized Linearizations For Rectangular Matrix Polynomials, 2019 Indian Institute of Technology Guwahati

#### Vector Spaces Of Generalized Linearizations For Rectangular Matrix Polynomials, Biswajit Das, Shreemayee Bora

*Electronic Journal of Linear Algebra*

The complete eigenvalue problem associated with a rectangular matrix polynomial is typically solved via the technique of linearization. This work introduces the concept of generalized linearizations of rectangular matrix polynomials. For a given rectangular matrix polynomial, it also proposes vector spaces of rectangular matrix pencils with the property that almost every pencil is a generalized linearization of the matrix polynomial which can then be used to solve the complete eigenvalue problem associated with the polynomial. The properties of these vector spaces are similar to those introduced in the literature for square matrix polynomials and in fact coincide with them when ...

Describing Convex Polyhedral Growth, 2019 SUNY Geneseo

#### Describing Convex Polyhedral Growth, Bethanna Jones

*Papers, Posters, and Recordings*

We outline a method of describing convex polyhedra in computer graphics, using both algorithmic and mathematical models. Each polyhedron is described in our method with a unique set of planes that define the polyhedron’s faces. These planes thereby define the polyhedron’s shape, extent, and orientation. This definition allows for a computationally easy discrete growth process in which a polyhedron grows in directions perpendicular to some or all of its faces, preserving orientation and morphology. In future work, this method can potentially be applied to the realistic modeling of individual crystal growth and group aggregation on a substrate.

Generating Spectra Using Pca-Based Spectral Mixture Models, 2019 Olivet Nazarene University

#### Generating Spectra Using Pca-Based Spectral Mixture Models, Joseph S. Makarewicz, Heather D. Makarewicz

*Scholar Week 2016 - present*

PCA-based spectra mixture models have been created for several laboratory mixture data sets. This presentation provides examples of spectra that were generated using PCA-based spectra mixture models.

A More Powerful Unconditional Exact Test Of Homogeneity For 2 × C Contingency Table Analysis, 2019 Chapman University

#### A More Powerful Unconditional Exact Test Of Homogeneity For 2 × C Contingency Table Analysis, Louis Ehwerhemuepha, Heng Sok, Cyril Rakovski

*Mathematics, Physics, and Computer Science Faculty Articles and Research*

The classical unconditional exact *p*-value test can be used to compare two multinomial distributions with small samples. This general hypothesis requires parameter estimation under the null which makes the test severely conservative. Similar property has been observed for Fisher's exact test with Barnard and Boschloo providing distinct adjustments that produce more powerful testing approaches. In this study, we develop a novel adjustment for the conservativeness of the unconditional multinomial exact *p*-value test that produces nominal type I error rate and increased power in comparison to all alternative approaches. We used a large simulation study to empirically estimate ...

Algorithms For Bohemian Matrices, 2019 The University of Western Ontario

#### Algorithms For Bohemian Matrices, Steven E. Thornton

*Electronic Thesis and Dissertation Repository*

This thesis develops several algorithms for working with matrices whose entries are multivariate polynomials in a set of parameters. Such parametric linear systems often appear in biology and engineering applications where the parameters represent physical properties of the system. Some computations on parametric matrices, such as the rank and Jordan canonical form, are discontinuous in the parameter values. Understanding where these discontinuities occur provides a greater understanding of the underlying system.

Algorithms for computing a complete case discussion of the rank, Zigzag form, and the Jordan canonical form of parametric matrices are presented. These algorithms use the theory of regular ...

The Mathematical Modeling Of Ballet, 2019 Louisiana Tech University

#### The Mathematical Modeling Of Ballet, Kendall Gibson

*Mathematics Senior Capstone Papers*

This project aims to analyze the connections between ballet and mathematics. Specifically, this project focuses on analyzing the three-dimensional surfaces created as a dancer performs ballet choreography. The primary goal is to use a Vicon motion capture system in conjunction with MATLAB to model the three-dimensional lines and surfaces created by a dancer’s legs as she performs specific ballet movements. The movements used for this experiment were a pique turn and a rond de jambe. The data was collected using sensors to create objects in Vicon to record the position of the ankle, knee, and hip of the working ...

Optimal Conditional Expectation At The Video Poker Game Jacks Or Better, 2019 University of Utah

#### Optimal Conditional Expectation At The Video Poker Game Jacks Or Better, Stewart N. Ethier, John J. Kim, Jiyeon Lee

*UNLV Gaming Research & Review Journal*

There are 134,459 distinct initial hands at the video poker game Jacks or Better, taking suit exchangeability into account. A computer program can determine the optimal strategy (i.e., which cards to hold) for each such hand, but a complete list of these strategies would require a book-length manuscript. Instead, a hand-rank table, which fits on a single page and reproduces the optimal strategy perfectly, was found for Jacks or Better as early as the mid 1990s. Is there a systematic way to derive such a hand-rank table? We show that there is indeed, and it involves finding the ...

Neural Machine Translation, 2019 Loyola Marymount University

#### Neural Machine Translation, Quinn M. Lanners, Thomas Laurent

*Honors Thesis*

Neural Machine Translation is the primary algorithm used in industry to perform machine translation. This state-of-the-art algorithm is an application of deep learning in which massive datasets of translated sentences are used to train a model capable of translating between any two languages. The architecture behind neural machine translation is composed of two recurrent neural networks used together in tandem to create an Encoder Decoder structure. Attention mechanisms have recently been developed to further increase the accuracy of these models. In this senior thesis, the various parts of Neural Machine Translation are explored towards the eventual creation of a tutorial ...

Heterogeneous Boolean Networks With Two Totalistic Rules, 2019 University of Nebraska at Omaha

#### Heterogeneous Boolean Networks With Two Totalistic Rules, Katherine Toh

*Student Research and Creative Activity Fair*

Boolean Networks are being used to analyze models in biology, economics, social sciences, and many other areas. These models simplify the reality by assuming that each element in the network can take on only two possible values, such as ON and OFF. The node evolution is governed by its interaction with other nodes in its neighborhood, which is described mathematically by a Boolean function or rule. For simplicity reasons, many models assume that all nodes follow the same Boolean rule. However, real networks often use more than one Boolean rule and therefore are heterogeneous networks. Heterogeneous networks have not yet ...

Parametric Natura Morta, 2019 Independent researcher, Palermo, Italy

#### Parametric Natura Morta, Maria C. Mannone

*The STEAM Journal*

Parametric equations can also be used to draw fruits, shells, and a cornucopia of a mathematical still life. Simple mathematics allows the creation of a variety of shapes and visual artworks, and it can also constitute a pedagogical tool for students.