# Papers

#### Field Simulation and Learning

The focus of this paper is on modelling 2D fields across time. The Gaussian Markov Random Field (GMRF) approach is sometimes used to model stationary (not time dependent) 2D fields. The GMRF approach is extended with analytic partial differential equation techniques to accommodate time dependent fields. This work was done at the Technical University of Hamburg in Germany, sponsored by the DAAD.

#### The Prime Number Theorem

As part of an independent study at the University of Rochester I reconstructed the proof of the Prime Number Theorem. I tried to make this proof very instructional and informative, leaving few details to the imagination. Have fun, this one is a YOWZA!

#### Reinforcement Learning in Two Player Games

This work combines ideas from traditional reinforcement learning, game theory, and meta learning. It is a largely instructional paper, introducing the basic notions of Markov Decision Processes, Q-Learning, and Nash Equilibria, but quickly moves on to some novel algorithms that I developed during my REU at the University of Rochester.

#### The Shallow Water Equations

This report was written for the Applied Boundary Value Problems course at University of Rochester. It was written with Nicholas Baronowsky and Sebastian Jakymiw. The report explores the Shallow Water Equations.

#### Snowboarding the Halfpipe with Classical Mechanics

This paper combines a spring mechanical model for motion of a snowboarder in the halfpipe with a rotating projectile motion model for the snowboarder in the air. This model is used to explore techniques that the snowboarder can use to maximize his score in competitions.

#### On Gordeev's "On P versus NP"

In the paper "On P versus NP," Lev Gordeev attempts to extend the method of approximation, which successfully proved exponential lower bounds for monotone circuits, to the case of De Morgan Normal (DMN) circuits. As in Razborov's proof of exponential lower bounds for monotone circuits, Gordeev's work is focused on the NP-complete problem CLIQUE. If successful in proving exponential DMN circuit lower bounds for CLIQUE, Gordeev would prove that P ≠ NP. However, we show that Gordeev makes a crucial mistake in Lemma 12. This mistake comes from only approximating operations over positive circuit inputs. Furthermore, we argue that efforts to extend the method of approximation to DMN circuits will need to approximate negated inputs as well.