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.

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.

 Solar power provides a renewable energy resource that reduces carbon footprints and lowers global warming. Solar panels use photovoltaics which convert light to electricity. Most commercial solar panels use silicon wafers. Electrons in these semiconducting silicon panels are freed by solar energy and are induced to travel through an electrical circuit, powering electrical devices or sending electricity to the grid. We have analyzed the reported crystal structure of silicon, which crystallizes in the same pattern as diamond and has a face centered cubic structure with lattice constant 5.4307 Å. We employed vector calculus-based methods to calculate the nearest-neighbor bond lengths (2.3516 Å) and bond angles (109.471^o) of crystalline silicon. These calculated bond-lengths and angle values are in good agreement with reported data. We visualized the electronic charge-density of silicon. Using vector-calculus based methods, we derived the equation for the plane of the solar panel and estimated the power that a solar panel can produce. Real time data from solar panel grids are currently available from energy databases. We determined the total energy produced by a solar panel array over the course of a day by finding the area under the power-vs-time real-time data reported in energy databases using integral calculus-based methods. To understand seasonal variations, we compared solar energy production on a hot summer day and during an overcast winter day. Our studies provide a microscopic atomic level understanding of solar energy and provides an integrated study of mathematics with solar physics and engineering.

A graph is a collection of vertices and edges. In computer science, graphs are often called networks and form the basis for many data structures and search algorithms. A matching of a graph is a selection of edges that share no common endpoints. The set of all matchings of a graph forms a simplicial complex which we call the matching complex. We are interested in the relationship between a graph and its matching complex and have been exploring whether we can characterize all simplicial complexes that are matching complexes. What structure does the matching complex imply about the graph and vice versa? We have also been interested in connections between matching complexes and well-known simplicial complexes– in particular, two-dimensional Buchsbaum complexes. These have much more structure than simplicial complexes in general, so they lend themselves to interesting questions. How can we characterize all two-dimensional Buchsbaum complexes that are matching complexes? We have also developed an interest in sequences of graphs generated by taking repeated matching complexes. Understanding these sequences would allow us to categorize graphs using the matching operation. Which graphs go to the empty set after a finite number of iterated matchings? For what finite values does this occur? What common structure do these graphs have? Investigating these questions will allow us to categorize graphs and complexes using the matching operation. We hope to make connections between graphs or categories of graphs that would otherwise remain disconnected. In pursuing these queries not only are we seeking answers but a development of tools which can be applied to further exploration.