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.

Data encryption finds important applications in cybersecurity and is vital for sensitive data including online financial transactions, preventing data breach from social media platforms, data security, etc. We have used various mathematical algorithms using matrix algebra for data encryption. We have developed a phone app for secure data transmission and relay which is suitable for data encryption for email, internet and social media. We have used static and dynamic data encryption as well as data scrambling methods to provide additional layer of security. The methods we use are suitable for storage and transmission of text, images, audio and video on the internet. The algorithms we employ include Hill Cipher, Modulo arithmetic, hash functions, random data shuffling, data scrambling, LU factorization and other linear algebra methods for data encryption. We have studied advanced encryption standards (AES) used for compliance for financial processing. We propose mathematical algorithms involving end-to-end data encryption which may be suitable for video data relay or online data processing for banking, credit card and other financial transactions. The project provides real-world applications of Mathematics for Cybersecurity and Data Sciences.

 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.