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.

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.

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.

There is currently extensive demand for optical media like CDROM, DVD and Blue ray 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 blue ray 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 Blue ray disks. The advancement from CDs to DVDs yields higher data storage densities. In the high capacity blue rays 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 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.