Regularity of weak solutions of a complex Monge-Amp\`ere equation

We prove the smoothness of weak solutions to an elliptic complex Monge-Ampere equation, using the smoothing property of the corresponding parabolic flow.Comment: 9 pages; removed a sentence from the introduction, other minor change

VIRTUAL ROBOT LOCOMOTION ON VARIABLE TERRAIN WITH ADVERSARIAL REINFORCEMENT LEARNING

Reinforcement Learning (RL) is a machine learning technique where an agent learns to perform a complex action by going through a repeated process of trial and error to maximize a well-defined reward function. This form of learning has found applications in robot locomotion where it has been used to teach robots to traverse complex terrain. While RL algorithms may work well in training robot locomotion, they tend to not generalize well when the agent is brought into an environment that it has never encountered before. Possible solutions from the literature include training a destabilizing adversary alongside the locomotive learning agent. The destabilizing adversary aims to destabilize the agent by applying external forces to it, which may help the locomotive agent learn to deal with unexpected scenarios. For this project, we will train a robust, simulated quadruped robot to traverse a variable terrain. We compare and analyze Proximal Policy Optimization (PPO) with and without the use of an adversarial agent, and determine which use of PPO produces the best results

Note on resonance varieties

We study the irreducibility of resonance varieties of graded rings over an exterior algebra E with particular attention to Orlik-Solomon algebras. We prove that for a stable monomial ideal in E the first resonance variety is irreducible. If J is an Orlik- Solomon ideal of an essential central hyperplane arrangement, then we show that its first resonance variety is irreducible if and only if the subideal of J generated by all degree 2 elements has a 2-linear resolution. As an application we characterize those hyperplane arrangements of rank less than or equal to 3 where J is componentwise linear. Higher resonance varieties are also considered. We prove results supporting a conjecture of Schenck-Suciu relating the Betti numbers of the linear strand of J and its first resonance variety. A counter example is constructed that this conjecture is not true for arbitrary graded ideals

The Monge-Amp\`ere operator and geodesics in the space of K\"ahler potentials

It is shown that geodesics in the space of K\"ahler potentials can be uniformly approximated by geodesics in the spaces of Bergman metrics. Two important tools in the proof are the Tian-Yau-Zelditch approximation theorem for K\"ahler potentials and the pluripotential theory of Bedford-Taylor, suitably adapted to K\"ahler manifolds.Comment: 25 pages, no figure, minor misprints correcte

Compositional Distributional Semantics with Long Short Term Memory

We are proposing an extension of the recursive neural network that makes use of a variant of the long short-term memory architecture. The extension allows information low in parse trees to be stored in a memory register (the `memory cell') and used much later higher up in the parse tree. This provides a solution to the vanishing gradient problem and allows the network to capture long range dependencies. Experimental results show that our composition outperformed the traditional neural-network composition on the Stanford Sentiment Treebank.Comment: 10 pages, 7 figure