High-order terms in the renormalized perturbation theory for the Anderson impurity model

We study the renormalized perturbation theory of the single-impurity Anderson model, particularly the high-order terms in the expansion of the self-energy in powers of the renormalized coupling $\tilde{U}$. Though the presence of counter-terms in the renormalized theory may appear to complicate the diagrammatics, we show how these can be seamlessly accommodated by carrying out the calculation order-by-order in terms of skeleton diagrams. We describe how the diagrams pertinent to the renormalized self-energy and four-vertex can be automatically generated, translated into integrals and numerically integrated. To maximize the efficiency of our approach we introduce a generalized $k$-particle/hole propagator, which is used to analytically simplify the resultant integrals and reduce the dimensionality of the integration. We present results for the self-energy and spectral density to fifth order in $\tilde{U}$, for various values of the model asymmetry, and compare them to a Numerical Renormalization Group calculation.Comment: 11 pages, 8 figure

Balancing Selection Pressures, Multiple Objectives, and Neural Modularity to Coevolve Cooperative Agent Behavior

Previous research using evolutionary computation in Multi-Agent Systems indicates that assigning fitness based on team vs.\ individual behavior has a strong impact on the ability of evolved teams of artificial agents to exhibit teamwork in challenging tasks. However, such research only made use of single-objective evolution. In contrast, when a multiobjective evolutionary algorithm is used, populations can be subject to individual-level objectives, team-level objectives, or combinations of the two. This paper explores the performance of cooperatively coevolved teams of agents controlled by artificial neural networks subject to these types of objectives. Specifically, predator agents are evolved to capture scripted prey agents in a torus-shaped grid world. Because of the tension between individual and team behaviors, multiple modes of behavior can be useful, and thus the effect of modular neural networks is also explored. Results demonstrate that fitness rewarding individual behavior is superior to fitness rewarding team behavior, despite being applied to a cooperative task. However, the use of networks with multiple modules allows predators to discover intelligent behavior, regardless of which type of objectives are used