Mathematics Commons^™

Sinusoidal Projection For 360° Image Compression And Triangular Discrete Cosine Transform Impact In The Jpeg Pipeline, Iker Vazquez Lopez

Boise State University Theses and Dissertations

The equirectangular projection is commonly used to store and transmit 360' images. However, using the equirectangular projection to store and transmit 360° images is not efficient due to its natural topographic redundancy. To generate the 360° image, captured pixels that form a spherical point cloud in the 3D space are projected onto a 2D plane using the equirectangular projection; generating redundant pixels in the process. These extra pixels in the image add extra memory requirements that have low impact in the final image quality. This dissertation presents results of research into the compression of 360° spherical imagery. It examines and …

Fantasy On A Baseball Seam, Genevieve Ahlstrom, Thomas Q. Sibley

Mathematics Faculty Publications

No abstract provided.

Tcr-L: An Analysis Tool For Evaluating The Association Between The T-Cell Receptor Repertoire And Clinical Phenotypes, Meiling Liu, Juna Goo, Yang Liu, Wei Sun, Michael C. Wu, Li Hsu, Qianchuan He

Mathematics Faculty Publications and Presentations

Background: T cell receptors (TCRs) play critical roles in adaptive immune responses, and recent advances in genome technology have made it possible to examine the T cell receptor (TCR) repertoire at the individual sequence level. The analysis of the TCR repertoire with respect to clinical phenotypes can yield novel insights into the etiology and progression of immune-mediated diseases. However, methods for association analysis of the TCR repertoire have not been well developed.

Methods: We introduce an analysis tool, TCR-L, for evaluating the association between the TCR repertoire and disease outcomes. Our approach is developed under a mixed effect modeling, where …

Topological Data Analysis And Ant Interaction Networks, Adam Banatwala, Esther Rønn

Mathematics & Computer Science Student Scholarship

Adam Banatwala ’22, Majors: Mathematics and Finance
Esther Rønn ’23, Majors: Physics and Mathematics
Faculty Mentor: Dr. Laura Murray, Mathematics and Computer Science

Our research group used topological data analysis (TDA) to quantify the movement and behavior of ants in a colony.

We extracted higher dimensional networks from point cloud data collected from Dr. James Waters’ lab. Varying the proximity parameter in this construction gives a sequence of networks. We analyzed the enduring topological features of these networks, and how these features evolve over time as the ants move in the colony. Both the experimental and null model simulation data …

Efficient Handover Mechanisms For Heterogeneous Networks., Shankar Kumar Ghosh Dr.

Doctoral Theses

In this thesis, some analytical frameworks have been developed to analyze the effect of different system parameters on handover performances in heterogeneous network (HetNet) and based on such frameworks, some efficient handover algorithms have been proposed. The study starts with an analytical framework to investigate the effect of resource allocation mechanisms, upper layer mobility management protocols (MMPs) and handover decision metrics on user perceived throughput. This analysis reveals that among other factors, handover decision metric plays a crucial role in determining user perceived throughput in HetNet. Subsequently, we develop two handover decision metrics for ultra dense networks (UDN) and unlicensed …

A Computationally Efficient Clustering Linear Combination Approach To Jointly Analyze Multiple Phenotypes For Gwas, Meida Wang, Shuanglin Zhang, Qiuying Sha

Michigan Tech Publications, Part 1

There has been an increasing interest in joint analysis of multiple phenotypes in genome-wide association studies (GWAS) because jointly analyzing multiple phenotypes may increase statistical power to detect genetic variants associated with complex diseases or traits. Recently, many statistical methods have been developed for joint analysis of multiple phenotypes in genetic association studies, including the Clustering Linear Combination (CLC) method. The CLC method works particularly well with phenotypes that have natural groupings, but due to the unknown number of clusters for a given data, the final test statistic of CLC method is the minimum p-value among all p-values of the …

How To Guard An Art Gallery: A Simple Mathematical Problem, Natalie Petruzelli

The Review: A Journal of Undergraduate Student Research

The art gallery problem is a geometry question that seeks to find the minimum number of guards necessary to guard an art gallery based on the qualities of the museum’s shape, specifically the number of walls. Solved by Václav Chvátal in 1975, the resulting Art Gallery Theorem dictates that ⌊n/3⌋ guards are always sufficient and sometimes necessary to guard an art gallery with n walls. This theorem, along with the argument that proves it, are accessible and interesting results even to one with little to no mathematical knowledge, introducing readers to common concepts in both geometry and graph …

Classifications Of Transformations Of Km#Tn And Pm#Tn Via Symmetry Groups, Adam Banatwala, Jensen Barry, Mackenzie Maude

Mathematics & Computer Science Student Scholarship

Adam Banatwala ’22, Majors: Mathematics and Finance
Jensen Barry ’22, Majors: Biology and Mathematics
Mackenzie Maude ’22, Majors: Mathematics and Art History
Faculty Mentor: Dr. C. Joanna Su, Mathematics and Computer Science

One of the main topics in topology is the classification and comparison of shapes and surfaces. Since Spring 2020, our research group has been using symmetry groups to classify the 1- and 2-dimensional orientable and non-orientable closed surfaces.

First, the group worked on the symmetry groups of the 1- and 2-dimensional orientable closed surfaces; namely, the classification on V?? 1 (the one-point adjoint of n circles) …

Meas: Exploring Links Between Implementation And Standards Mastery, Noah Silver

Honors Projects

In order to effectively enhance a student’s mathematical understanding and development in the field of mathematics, students need to engage in problem solving. Model eliciting activities, or MEAs, provide students with tasks that promote higher level thinking and the ability to utilize mathematics outside of the classroom; they also align and promote the utilization of the Common Core State Standards and Standards for Mathematical Practice. Research suggests that the language and motivation promoted by MEAs enriches engagement and increases student ability and performance of traditional and real-world mathematics. Use of technology further supports these goals. Through the analysis of checkpoint …

Sharp Inequalities Of The X-Ray Transform And The Competing Symmetries Argument, Arthur D. Dressenwall

Mathematics, Statistics, and Computer Science Honors Projects

We examine the $k=1$ case of a conjecture by Baernstein and Loss pertaining to the operator norm of the $k$-plane transform from $L^p(\R^d)$ space to $L^q(\M)$ space. Previous work on this problem by Carlen and Loss, as well as by Drouot, has used an iterative technique known as the ``competing symmetries argument’’ to prove this conjecture in the $q=2$ and $q=d+1$ cases. We summarize the conjecture and this proof technique, then perform work that strongly suggest that no sufficiently ``nice” transformation exists that can be used to apply the competing symmetries argument to other cases of the conjecture.

Bounding Projective Hypersurface Singularities, Benjamin Castor

Arts & Sciences Graduate Student Theses and Dissertations

We first provide background necessary for understanding monodromy and spectra. We then compare several different methods involving Hodge-theoretic spectra of singularities which produce constraints on the number and type of isolated singularities on a projective hypersurface of fixed degree. In particular, we introduce a method based on the spectrum of the nonisolated singularity at the origin of the affine cone on such a hypersurface, and relate the resulting explicit formula to Varchenko’s bound. We then provide a purely combinatorial interpretation of our theorems and our conjecture.

A New Method To Compute The Hadamard Product Of Two Rational Functions, Ishan Kar

Rose-Hulman Undergraduate Mathematics Journal

The Hadamard product (denoted by∗) of two power series A(x) =a₀+a₁x+a₂x²+···and B(x) =b₀+b₁x+b₂x²+··· is the power series A(x)∗B(x) =a₀b₀+a₁b₁x+a₂b₂x²+···. Although it is well known that the Hadamard product of two rational functions is also rational, a closed form expression of the Hadamard product of rational functions has not been found. Since any rational power series can be expanded by partial fractions as a polynomial plus a sum of power series …

Unknowable Truths: The Incompleteness Theorems And The Rise Of Modernism, Caroline Tvardy

Honors Scholars Collaborative Projects

This thesis evaluates the function of the current history of mathematics methodologies and explores ways in which historiographical methodologies could be successfully implemented in the field. Traditional approaches to the history of mathematics often lack either an accurate portrayal of the social and cultural influences of the time, or they lack an effective usage of mathematics discussed. This paper applies a holistic methodology in a case study of Kurt Gödel’s influential work in logic during the Interwar period and the parallel rise of intellectual modernism. In doing so, the proofs for Gödel’s Completeness and Incompleteness theorems will be discussed as …

Lines On A Smooth Projective Surface, Jordan Demoura

Mathematics & Computer Science Student Scholarship

Jordan DeMoura ’22
Major: Mathematics
Faculty Mentor: Dr. Su-Jeong Kang, Math

This research is to investigate lines on a smooth projective surface. A quadric surface contains two families of planes that provide a ruling of the surface. A cubic surface contains twenty-seven lines, and we provide a complete description of these lines for a Fermat cubic surface. Furthermore, under the Plucker embedding, we show that each family of the lines on a quadric surface corresponds to plane conic curves lying on complementary planes in the projective space of dimension five.

Quantization For Infinite Affine Transformations, Dogan Comez, Mrinal Kanti Roychowdhury

School of Mathematical & Statistical Sciences Faculty Publications

Quantization for a probability distribution refers to the idea of estimating a given probability by a discrete probability supported by a finite set. In this article, we consider a probability distribution generated by an infinite system of affine transformations {S-ij} on R-2 with associated probabilities {p(ij)} such that p(ij) > 0 for all i, j is an element of N and Sigma(infinity)(i,j=1) p(ij) = 1. For such a probability measure P, the optimal sets of n-means and the nth quantization error are calculated for every natural number n. It is shown that the distribution of such a probability measure is the …

A Study Of Semiprime Arithmetic Sequences, Sam C. Uselton

Honors Scholars Collaborative Projects

Semiprimes are positive integers of the form n = p*q where p and q are primes that are not necessarily distinct. Semiprimes can occur in arithmetic sequence, and the upper bound changes depending on the distance, d, between semiprimes in the sequence. The goal of this work is to present and prove theorems for different cases of semiprime arithmetic sequences, including d = 2k+1 where k is greater than or equal to 0 and d=2k where k is positive and not a multiple of 3, as well as a general case for finding upper bounds for varying distances. The building …

A Novel Tcr Clustering Method For Sars-Cov-2 Epitopes, Naziba A. Nuha

Mathematics Colloquium Series

T-cell epitopes are peptides generated from antigens that are presented by MHC class I and class II molecules to T-cells. These epitopes are usually identified by T-cell receptors (TCRs) of CD4 T-cells which then causes transformation of CD4 T-cells to helper or regulatory T cells. Recently, there has been growing interest in the role of T cells and their involvement in various ailments including SARS-COV-2, cancer, autoimmune diseases and other infectious diseases. However, the mechanism of TCR epitope recognition by Tcell receptors (TCRs) of CD4 T-cells at a repertoire level is still not fully understood. In this project, we reviewed …

A Strange Attractor Of Primes, Alexander Hare

ONU Student Research Colloquium

The greatest prime factor sequences (GPF sequences), born at ONU in 2005, are integer sequences satisfying recursions in which every term is the greatest prime factor of a linear combination with positive integer coefficients of the preceding k terms (where k is the order of the sequence), possibly including a positive constant. The very first GPF sequence that was introduced satisfies the recursion x(n+1)=GPF(2*x(n)+1). In 2005 it was conjectured that no matter the seed, this particular GPF sequence enters the limit cycle (attractor) 3-7-5-11-23-47-19-13. In our current work, of a computational nature, we introduce the functions “depth” – where depth(n) …

Artsy Chaos: The Secret Life Of A Class Of Trigonometric Sums, Kaleb Swieringa, Joelena Brown, Rachael Harbaugh, Francis Nadolny

ONU Student Research Colloquium

We start from classical trigonometric sums (of terms such as k^n*cos(k), k^n*sin(k) - where n is a positive integer). These classical sums allow fairly straightforward closed form representations. In our work we considered changing the arguments of the trigonometric factors to powers (so that they get replaced by cos(k^a) and sin(k^a) - for a positive real exponent that may or may not be an integer), while also introducing in any term of such a sum a "rotational" factor of the form omega^k, where "omega" is a complex number of modulus 1 (that may or may not be a root of …

Constructing Digital Elevation Models From Lake Michigan Dune Imagery, Blake Harlow, Jack Krebsbach

21st Annual Celebration of Undergraduate Research and Creative Activity (2022)

Lake Michigan dune complexes evolve as winds and waves move the sand, causing major topographic changes over time. The Hope College Dune Group has been using remote sensing data from drone imagery to model various aspects of these dunes. In order to better understand the dune dynamics, a digital terrain model (DTM) is desired. A DTM is an elevation map of the dune’s bare ground surface which excludes ground obstructions such as trees and bushes. We have attempted to use traditional methods that have been developed for processing LIDAR data in order to construct these DTMs from our drone imagery, …