The prevalence of differential equations as a mathematical technique has refined the fields of control theory and constrained optimization due to the newfound ability to accurately model chaotic, unbalanced systems. However, in recent research, systems are increasingly more nonlinear and difficult to model using Differential Equations only. Thus, a newer technique is to use policy iteration and Reinforcement Learning, techniques that center around an action and reward sequence for a controller. Reinforcement Learning (RL) can be applied to control theory problems since a system can robustly apply RL in a dynamic environment such as the cartpole system (an inverted pendulum). This solution successfully avoids use of PID or other dynamics optimization systems, in favor of a more robust, reward-based control mechanism. This paper applies RL and Q-Learning to the classic cartpole problem, while also discussing the mathematical background and differential equations which are used to model the aforementioned system.

Increasingly, genomics is being used for the prediction of specific traits and diseases (phenotypes) among humans. Wider availability of genomics data through multiple research projects (such as International HapMap Project1 and 1000 Genomes2) has been a catalyst in that direction. With the recent advances in machine learning and big data analysis, data computation resources and data models needed for genomics data analysis are readily available. However, the prediction of traits and diseases has its own challenges in terms of computational requirements and computational analysis, statistical analysis (example: confounding variables), and limited quality of data collection. Linear Mixed Models (LMM, a type of linear regression) is a common approach for Genome-wide Association Studies (GWAS) for the prediction of common traits among humans using genomics. This paper researches the existing LMM-based approaches for Genome-wide Association Studies (GWAS), describes the experiment performed on FaST-LMM approach from Microsoft Research, and then proposes an enhanced approach (called LMM-22) on how to address computational and statistical issues. LMM-22 focuses on the parallelization of LMM computations and execution of LMM-22 on General Purpose Graphics Processing Units (GPU) as against CPUs to accelerate the LMM approach for GWAS studies.

We present a very computationally light and fast approximation algorithm and then verify it with genetic algorithm and simulated annealing. We show that our algorithm is on par with GA and SA in terms of output produced while having a tightly bounded time complexity. Our algorithm works best when there is a strong positive correlation between the reliability of a component and its cost. We present two algorithms with the same essence. One of them is system cost bounded and the other is target reliability bounded. Our proposed algorithm works on a subsystem level redundancy instead of component level redundancy.