By adapting a previously written percolation model in C, the threshold probabilities for square, triangular, and cubic lattice types were confirmed. An algorithm to count the distribution of cluster sizes at a variety of percolation probabilities was developed, and the expected trends towards the so called infinite cluster was achieved. An equivalent bond percolation model was adapted to the original site algorithm, and by treating occupied bonds as springs, a total compression trend for the model was constructed, which implied that structures under the boundary conditions that were imposed does not have behavior that changes the total compression constant significantly at the percolation threshold.\\

Pursuit and evasion tends to be incorporated in human nature from a very long time with a very huge range of activities. Here, we are going to create an intelligent chase algorithm, which uses two basic approaches. After which we will use the newly created equations to simulate both approaches and provide graphical results. This analysis is based upon the fact that in modern days we can estimate the speed of a moving object and then chase it down depending upon its speed. We will also be taking the accuracy of such estimation in picture and depending upon a certain accuracy and other inputs the simulation will provide the result and performance of both the approaches.

In this paper we discuss how to price American, European and Asian options using a geometric Brownian motion model for stock price. We investigate the analytic solution for Black-Scholes differential equation for European options and consider numerical methods for approximating the price of other types of options. These numerical methods include Monte Carlo, binomial trees, trinomial trees and finite difference methods. We conclude our discussion with an investigation of how these methods perform with respect to the changes in different Greeks. Further analysing how the value of a certain Greeks affect the price of a given option.