Analytical expressions for the polarizability of the honeycomb lattice

We present analytical expressions for the polarizability $P_\mu(q_x,\omega)$ of graphene modeled by the hexagonal tight-binding model for small wave number $q_x$, but arbitrary chemical potential $\mu$. Generally, we find $P_\mu(q_x,\omega)=P_\mu^(\omega)$ with $\omega_q=v_Fq_x$ the Dirac energy, where the first term is due to intra-band and the second due to inter-band transitions. Explicitly, we derive the analytical expression for the imaginary part of the polarizability including intra-band contributions and recover the result obtained from the Dirac cone approximation for $\mu\rightarrow0$. For $\mu<\sqrt{3}t$, there is a square-root singularity at $\omega_q=v_Fq_x$ independent of $\mu$. For doping levels close to the van Hove singularity, $\mu=t\pm\delta\mu$, $ImP_\mu(q_x,\omega)$ is constant for $\delta\mu/t<\omega/\omega_q\ll1$.Comment: 5 pages, 3 figures, 1 tabl

A Note on Kuhn's Theorem with Ambiguity Averse Players

Kuhn's Theorem shows that extensive games with perfect recall can equivalently be analyzed using mixed or behavioral strategies, as long as players are expected utility maximizers. This note constructs an example that illustrate the limits of Kuhn's Theorem in an environment with ambiguity averse players who use maxmin decision rule and full Bayesian updating.Comment: 7 figure

Creating new rural development strategies : the role of nonprofits

Rural areas ; Rural development ; Nonprofit organizations

Why invest in rural America and how? : a critical public policy question for the 21st century

Significant portions of rural America are in trouble. For some parts of rural America, the slow slide to no longer being economically, socially, or politically viable is within sight. At the same time, without intending it, rural America appears headed for a land of the rich and the poor—a rural America of resorts and pockets of persistent poverty.> Yet current rural policies are designed for the past, not the future. In terms of public dollars committed, today’s rural policy focuses primarily on two areas—agriculture and manufacturing. Neither focus can meet the future needs of rural people and their communities.> In his paper presented at this year’s conference sponsored by the Center for the Study of Rural America, Exploring Policy Options for a New Rural America, Dr. Stauber discusses how a successful rural policy must be crafted with three key societal benefits in mind—the survival of the rural middle class, reducing concentrated rural poverty, and sustaining and improving the quality of the natural environment.Rural areas ; Rural development