Finding All Nash Equilibria of a Finite Game Using Polynomial Algebra

The set of Nash equilibria of a finite game is the set of nonnegative solutions to a system of polynomial equations. In this survey article we describe how to construct certain special games and explain how to find all the complex roots of the corresponding polynomial systems, including all the Nash equilibria. We then explain how to find all the complex roots of the polynomial systems for arbitrary generic games, by polyhedral homotopy continuation starting from the solutions to the specially constructed games. We describe the use of Groebner bases to solve these polynomial systems and to learn geometric information about how the solution set varies with the payoff functions. Finally, we review the use of the Gambit software package to find all Nash equilibria of a finite game.Comment: Invited contribution to Journal of Economic Theory; includes color figure

Thermoelectric properties of Bi2Te3 atomic quintuple thin films

Motivated by recent experimental realizations of quintuple atomic layer films of Bi2Te3,the thermoelectric figure of merit, ZT, of the quintuple layer is calculated and found to increase by a factor of 10 (ZT = 7.2) compared to that of the bulk at room temperature. The large enhancement in ZT results from the change in the distribution of the valence band density of modes brought about by the quantum confinement in the thin film. The theoretical model uses ab initio electronic structure calculations (VASP) with full quantum-mechanical structure relaxation combined with a Landauer formalism for the linear-response transport coefficients.Comment: 4 figures, submitted to AP

Self-Repairing Codes for Distributed Storage - A Projective Geometric Construction

Self-Repairing Codes (SRC) are codes designed to suit the need of coding for distributed networked storage: they not only allow stored data to be recovered even in the presence of node failures, they also provide a repair mechanism where as little as two live nodes can be contacted to regenerate the data of a failed node. In this paper, we propose a new instance of self-repairing codes, based on constructions of spreads coming from projective geometry. We study some of their properties to demonstrate the suitability of these codes for distributed networked storage.Comment: 5 pages, 2 figure

From scattering theory to complex wave dynamics in non-hermitian PT-symmetric resonators

I review how methods from mesoscopic physics can be applied to describe the multiple wave scattering and complex wave dynamics in non-hermitian PT-symmetric resonators, where an absorbing region is coupled symmetrically to an amplifying region. Scattering theory serves as a convenient tool to classify the symmetries beyond the single-channel case and leads to effective descriptions which can be formulated in the energy domain (via Hamiltonians) and in the time domain (via time evolution operators). These models can then be used to identify the mesoscopic time and energy scales which govern the spectral transition from real to complex eigenvalues. The possible presence of magneto-optical effects (a finite vector potential) in multichannel systems leads to a variant (termed PTT' symmetry) which imposes the same spectral constraints as PT symmetry. I also provide multichannel versions of generalized flux-conservation laws.Comment: 10 pages, 5 figures, minireview for a theme issue, Philosophical Transactions of the Royal Society

Energy transport and fluctuations in small conductors

The Landauer-B\"uttiker formalism provides a simple and insightful way for investigating many phenomena in mesoscopic physics. By this approach we derive general formulas for the energy properties and apply them to the basic setups. Of particular interest are the noise properties. We show that energy current fluctuations can be induced by zero-point fluctuations and we discuss the implications of this result.Comment: Revised and corrected versio

Tunnel Magnetoresistance of a Single-Molecule Junction

Based on the non-equilibrium Green's function (NEGF) technique and the Landauer-B\"{u}ttiker theory, the possibility of a molecular spin-electronic device, which consists of a single C$_{60}$ molecule attached to two ferromagnetic electrodes with finite cross sections, is investigated. By studying the coherent spin-dependent transport through the energy levels of the molecule, it is shown that the tunnel magnetoresistance (TMR) of the molecular junction depends on the applied voltages and the number of contact points between the device electrodes and the molecule. The TMR values more than 60% are obtained by adjusting the related parameters.Comment: 5 pages, 3 figure