Undergraduate Research

University of Washington, Undergraduate Research Symposium, 2020, Applied Mathematics and Data Modeling

Datasets Across Disciplines: Setting the Groundwork for Universal Atomic Machine Learning

Presenters

• Chandler Joseph King, Sophomore, Pre-Major (Arts & Sciences)
• Kyle Jonson, Senior, Computer Science

Mentors

• Mehmet Sarikaya, Materials Science & Engineering
• Oliver Nakano-Baker, Materials Science & Engineering
• Siddharth Rath (rathsidd@uw.edu)

Abstract

Our goal is to predictively engineer bio/nanomaterial hybrid systems with targeted functionality in a wide range of practical, technical, and medical applications. The open literature provides datasets of the functional properties of crystals, aqueous chemicals, and biological macromolecules, but the design of hybrid systems necessitates the modeling of all of these molecular species in a single common framework. Molecular graph convolutional networks and other deep learning methods are capable to train on datasets from multiple disciplines simultaneously, but in order to build these networks, a far-reaching data infrastructure is needed. We have created this infrastructure for three data sets: The Immune Epitope Database (IEDB) of MHC-I binding peptides, the Quantum-Machine.org QM9 dataset (QM9), and results extracted from the Materials Project. The IEDB provides binding affinities between biological macromolecules (peptide sequences in association with multiple MHC-I alleles); QM9 consists of 140,000 small organic molecules encoded as SMILES strings and 17 associated properties (including thermodynamic, energetic, geometric, and electronic information). The Materials Project dataset provides band gaps and formation energies for 70,000 crystal structures. We present a standardized train/test split and machine-learning-ready import interface for each of these datasets, as well as early results on co- and cross-training of deep neural networks across multiple datasets. The framework is expandable to new datasets and provides a strong foundation for ongoing efforts to build universal molecular encoding neural networks.

Euclid's Tutorials and Alternatives to Euclidean Geometry

Presenter

• Jesse Loi, Senior, Mathematics, Philosophy
• McNair Scholar

Mentor

• Conor Mayo-Wilson, Philosophy, University of Washington, Seattle

Abstract

In proofs, one must begin with definitions. However, a definition must designate something that exists, which offers some doubt for Euclidean geometry. Johann Heinrich Lambert, a mathematician, argues in the “Theory of Parallel Lines” for the necessity of Euclidean geometry. Lambert infers the necessity of Euclidean geometry from Euclid's tutorial-like postulates. For example, Euclid proves the existence of triangles with the postulate "to construct an equilateral triangle from a line segment". Because the postulate is a problem for the reader, to solve the postulate is a proof itself. It is a proof because, since the diagrams will be made by the reader, there is no room for flaw or deception on Euclid’s side. For example, suppose one wanted to show the existence of dolphins in the water. One could present the doubter with a picture of dolphins in the water. The doubter could question the authenticity of the picture. However, if the individual asked the doubter “go to the water”, then the doubter cannot question such an example. Euclid does not make such a strong claim about the existence of parallel lines. Euclid only claims that non-parallel lines will meet at some point on either one side of the intersection or the other. Successors of Euclid in later centuries have made attempts to provide arguments for the construction of parallel lines. I conduct a textual analysis of Katherine Dunlop's "Why Euclid's Geometry Brooked No Doubt" to explore attempts to prove the necessity of parallel lines while also referencing Euclid's "Elements" as another source text. My research also draws from later mathematicians on non-Euclidean geometry. As a result of my project, I investigate the validity of offering proofs to readers via preconstructed diagrams. In addition, I offer another approach to mathematics education given more activity-focused curriculum.

On Subsystems of Second-Order Arithmetic Induced by Combinatorial Results

Presenter

• Jordan Leoron Charles Brown, Senior, Mathematics (Comprehensive)
• McNair Scholar

Mentor

• James Morrow, Mathematics

Abstract

There are many ways to axiomatize arithmetic. It is natural to ask which axiomatizations place stronger conditions on the structure of arithmetic than others. The language of second-order arithmetic allows for the expression of nearly all results considered arithmetical. The basic axiomatic system expressed in the language of second-order arithmetic is called RCA_0, and the statements provable in it are essentially those that are 'computably' true. The strongest axiomatic system considered is all of second-order arithmetic, denoted Z_2. For any collection S of statements in the language of second-order arithmetic such that each statement in S is provable from the axioms of Z_2, we can consider the axiomatic system consisting of RCA_0 and S; such a system is called a subsystem of second-order arithmetic. The relative strengths of the systems thus obtained is the subject of much research. Here, we examine the strength of the systems obtained when S is a collection of combinatorial results. In particular, we wish to determine whether Hindman's theorem is provable in ACA_0. Hindman's theorem is the result that, in any finite coloring of the natural numbers, there is an infinite set such that any finite sum of elements in the set have the same color. It is stated quite differently from ACA_0, a subsystem of Z_2 that is a natural analogue of Peano arithmetic. However, it is known that ACA_0 can be proved from the subsystem consisting of RCA_0 and Hindman's Theorem. We aim to determine here whether the converse is true. This has implications for the proof-theoretic strength of other 'Ramsey-type' combinatorial results.

Optimizing data storage in optical and magnetic media: Mathematical modeling and photonics studies

Presenters

• Kylie Dillon, Sophomore, Computer science, Lake Wash Tech Coll
• Sam F. (Sam) Wolf, Junior, Computer Science & Software Engineering
• Taylour Mills, Junior, Aeronautics & Astronautics
• Jay Quedado, Junior, Computer Engineering (Bothell)
• Alana Yao, Fifth Year, Computer Science & Software Engineering

Mentors

• Narayani Choudhury, Computer Science & Engineering, Mathematics, Physics, Lake Washington Institute of Technology, Kirkland
• Hany Roufael, Engineering & Mathematics, Physics, Lake Washington Institute of Technology

Abstract

There is currently extensive demand for optical media like CDROM, DVD and Bluray disks for data storage with computer technologies. Here we combine mathematical modelling studies and photonic laser diffraction experiments to study the optimization of data storage in different types of optical media. Using calculus-based studies, we estimated the data storage capacities in these systems and calculated the CD, DVD and Bluray disk arc length and data storage linear densities. These are in good agreement with reported values. Using red, blue and green laser sources at our photonics lab, we conducted laser diffraction studies and estimated the line spacing of CDROM, DVD and Bluray disks. The advancement from CDs to DVDs yields higher data storage densities. In the high capacity bluray disks, because the physical structures called pits that store data on the disks become smaller, there are other challenges in realizing these smaller devices, which make it more expensive. The CD/DVD players' lasers operate at the diffraction limit resolution of light and provide maximum data capacity for their geometry. Magnetic media like floppy disks, hard disk and magnetic tapes are also used for computer data storage. We have estimated the maximum data storage capacity from magnetic floppy discs. We used curve fitting methods to analytically represent the magnetic read-back pulse as Lorentzian functions for data modeling. Our studies provide an integrated STEM learning of data storage in optical and magnetic media.

The Calculus of Clovers

Presenters

• Dave Edward (Dave) Diaz, Sophomore, Civil Engineering, Lake Wash Tech Coll
• Alana Yao, Fifth Year, Computer Science & Software Engineering
• Eric Trofimchik
• Matthew Capozzoli, Non-Matriculated, Engineering, Lake Wash Tech Coll

Mentor

• Narayani Choudhury, Engineering, Mathematics, Physics, Lake Washington Institute of Technology, Kirkland

Abstract

The clover-leaf is a fundamental shape that manifests often in nature. We have studied the calculus of the clover-leaf shape. We use multivariable calculus-based methods to estimate the average height of water in a clover-shaped swimming pool. Using double integration with polar coordinates, we find the areas of these shapes. The methods we use are very generic that elucidate the calculus of clovers. Such studies have many real-world applications as this shape is seen in leaves, flowers, tRNA, etc. tRNA (transfer ribonucleic acid) is a type of RNA molecule that helps decode a messenger RNA (mRNA) sequence into a protein. Clover leaf shapes are used in engineering design elements. The electronic d-orbitals have a three-dimensional clover leaf shape. The Chandra observatory discovered exciting findings of clover-leaf quasars that provides evidence of large-scale star formation in the early universe. We have a cloverleaf interchange at the 85th street at Kirkland. The calculus of clovers thus has many applications in fundamental sciences, engineering and transportation. We show how multivariable calculus studies using polar and cylindrical coordinates help study the characteristics of these shapes.

Mathematical Modelling and Design of a Robot prototype

Abdulrahman Ghalib, Sam Wolf, Geoffery Isom Powell, Narayani Choudhury

Robotics combines machining and artificial intelligence to create real world humanoid models for task automation and industrial applications. We have designed an in-house robot prototype having microprocessor controlled motion. The robot has lasers for eyes and has a position sensor with camera attached. We designed the gear box, track assembly and robot parts and have written software to control the motion of the robot. The robot is a good model for a Roomba-like vacuum cleaner. We created random walls using Monte Carlo simulations and used vector directed motion to control its motion for avoiding these random walls that the robot encounters to simulate real world experience. We have also studied robotic arm kinematics, using matrix algebra and trigonometry to help design a robot arm that we can rotate or translate to any point in three -dimensional space. We study both forward and reverse kinematics and have written software for the arm motion. Our studies provide an elegant educational platform for studies of robot motion along with simulating real-world experience.

Mathematical studies of data storage in CD ROM and DNA

Dylan Dean, Taylour Mills, Iuliia Dmitrieva, Narayani Choudhury

Current data storage elements have reached their threshold capabilities due to extensive data and limiting size requirements. Digital storage in DNA has aroused considerable interest as the next generation miniaturized high capacity storage device. DNA deoxyribonucleic acid forms the genetic blueprint of life and is the primary carrier of genetic information in living cells and organisms. Data storage in DNA involves encoding of digital binary data into synthesized DNA strands. Here, we employ calculus-based methods to provide a comparative study of data storage capacities of conventional CD ROM and DNA. We use parametric equations to model the spiral structure in CD ROM and double helix of DNA and employ calculus-based methods to study the arc length, curvature and topological properties of DNA. The data storage densities for binary, base 3 and base 4 in DNA are studied. The calculated data storage densities are found to be in good agreement with reported measurements. Recent studies demonstrate that magnetic nano-knots can be used for data storage. The topological properties of DNA including twists, links and knots thus provide additional attributes which may in future be used for data storage.

Network topology of knots and Borromean rings

Taylour Mills, John Hannon, Abdulrahman Ghalib, Narayani Choudhury

The use of magnetic nano-knots and Brunnian links for data storage and communications makes understanding the geometric and network topology  of knots and links very important. Recent reports suggest that DNA and other halogen networks self-assemble into exotic Borromean ring molecular topologies.  Borromean rings form a Brunnian link with three rings linked in such a way that no two alone are connected. Only when all the three rings come together does the linkage occur. Borromean links form the current logo of the International Mathematical Union and they display strength in unity. Understanding knots, links and their networking is central to our understanding of DNA, protein folding, polymers and other soft materials.

We have used a 3D printer to print and design a Borromean Math puzzle. The puzzle falls apart when a link is pulled out and is an excellent learning tool for studying Borromean link topologies.  We use mathematical methods using parametric equations to study Borromean rings and trefoil knots. We wrote computer visualization code using SAGE to display trefoil knots and complex Borromean links for circular, elliptical and other geometries. The SIEFERT surface of Borromean links are sketched using SeifertView and provide an aesthetic 3D view of the rings which can be oriented on a plane. The Seifert surface of a knot is a knot invariant; it is the  characteristic of the knot with the knot as a boundary. The adjacency matrix and topological connectivity of the links are studied using vector directed graph models.  A computer program is written to unravel the complex linking and intriguing connectivity properties of the trefoil knot and Borromean networks.

Vector Calculus studies of fullerenes

Iuliia Dmitrieva, Tom Skoczylas, Rami Manad, Narayani Choudhury
Lake Washington Institute of Technology

MATH Conference at Bellevue College, Feb. 19, 2018
UW Undergraduate Research conference, May, 2018

Polymer based fullerenes are used as photovoltaics in solar panels. Fullerene C60 molecules have
icosahedral based structures resembling geodesic domes. The mathematics of fullerenes have aroused much interest as their novel structures involve Golden ratios. Fullerenes have convex polyhedra which obey Euler’s topological rules. Here we use vector calculus methods to calculate bond lengths and bond angles and provide estimates for the volume of the molecule. We compare our results with reported numerical calculations of Van der Waals volume in fullerenes. To understand the critical effect of dimensionality on volume, we examine the volume of a hypersphere in n-dimensions. The project provides hands on exploration of real world problems and data visualization.

Multivariable calculus studies for environmental sciences and Engineering

Morgan Wolf, Sam Wolf, Narayani Choudhury
Lake Washington Institute of Technology

MATH Conference at Bellevue College, Feb. 19, 2018
UW Undergraduate Research conference, May, 2018

We explore applications of multivariable calculus for studying three dimensional wave media in our
environment. We use multivariable optimization methods to study the maxima and minima of three
dimensional functions. We use advanced data visualization and applications of divergence and curl to study ocean waves, including their divergence, curl, and circulation. Real world manifestations of scalar and vector fields in our environment are presented.

Monte Carlo simulations and the value of Pi

Sam Wolf, Narayani Choudhury
Lake Washington Institute of Technology

UW Undergraduate Research conference, May, 2018

Monte Carlo methods involve computational algorithms that rely on random sampling to obtain numerical results. Here we employ Monte Carlo simulations to estimate the numerical value of pi. The use of Monte Carlo methods for numerical integration and calculation of area and volumes of complex solids will be discussed. We will also discuss alternate numerical approaches including three dimensional Riemann sums and other strategies for derivation of volumes of complex solids.

Calculus-based Studies of Volume and Surface Area of Three-Dimensional Objects

Stephen Scharkov, Alexey Sinitsin, Narayani Choudhury
Lake Washington Institute of Technology

The Eleventh Annual Western Washington Community College Student Mathematics Conference Green River College, February 25, 2017

We derive parametric equations to model a three-dimensional flower vase. The volume and surface area of the vase is computed using calculus-based techniques employing cylindrical coordinates. Numerical multidimensional Riemann sums and regression-based methods are required for calculating the volume of the flower vase and objects having complex shapes. The project presents hands-on experience with data modeling, visualization and real life applications of calculus.

Vector Fields, Divergence, and Circulation

Aidan Hahn, Cesar Campos, Narayani Choudhury
Lake Washington Institute of Technology

The Eleventh Annual Western Washington Community College Student Mathematics Conference Green River College, February 25, 2017

We discuss the role of data visualization in elucidating the critical points of a multivariable function. The local and absolute maximum, minimum, and saddle points can be derived analytically using multivariable calculus. We employ surface plots, contour plots and vector gradient fields to study the local maxima and minima, divergence, curl, and circulation. The plots provide key insights into identifying whether a vector field is conservative or not. Quiver and Streamline integral plots provide visual solutions of differential equations. Data visualization enhances critical understanding and aids in the study of functions of many variables.

Robotic Arm Kinematics using Trigonometry and Matrix algebra

Sierra Bonilla, Narayani Choudhury
Lake Washington Institute of Technology

Math Conference at Edmonds Community College, Feb. 20, 2016

The kinematics for two and three dimensional robotic arms are studied using basic trigonometry. We use linear algebra to derive the transformation matrices for simple translations and rotations in three dimensions to derive the robotic arm coordinates for complex motions. Our studies use Matrix algebra and trigonometric methods effectively to study the kinematics and transformed coordinates of the robotic arm. A small computer code is being developed to derive the coordinates of the robotic arm for a range of motions. This research project serves as an elegant platform for hands on application of mathematical and computational methods for real life solutions.

Flightpath optimization and Dynamics of a Glider

Evvan Land, Tyler Goolsby, Lisa Camire, Narayani Choudhury
Lake Washington Institute of Technology

Math Conference at Edmonds Community College, Feb. 20, 2016

The flightpath of a hang glider is modeled using a polynomial function and its parameters optimized to minimize turbulence. We have also visualized the reported glider velocity data from University of Washington and analyzed it using calculus based methods to derive the instantaneous and average acceleration. This research project provides a novel avenue for hands on exploration and application of key mathematical concepts and data visualization.