With the rise in popularity of social media and e-commerce platforms, "discrete math" is at the heart of our online experiences, playing a front-end role in the recommendation of what to click, watch, or buy, or who to follow or friend, as well as a back-end role in data storage, access, and transfer. This thesis focuses on two such…

Weather forecasting consists of two processes, model integration and analysis (data assimilation). During the model integration, the state estimate produced by the analysis evolves to the next cycle time according to the atmospheric model to become the background estimate. The analysis then produces a new state estimate by combining the background…