Voting Power of Teams Working Together

Voting power determines the "power" of individuals who cast votes; their power is based on their ability to influence the winning-ness of a coalition. Usually each individual acts alone, casting either all or none of their votes and is equally likely to do either. This paper extends this standard "random voting" model to allow probabilistic voting, partial voting, and correlated team voting. We extend the standard Banzhaf metric to account for these cases; our generalization reduces to the standard metric under "random voting", This new paradigm allows us to answer questions such as "In the 2013 US Senate, how much more unified would the Republicans have to be in order to have the same power as the Democrats in attaining cloture?

Similarity Solutions of a Class of Perturbative Fokker-Planck Equation

In a previous work, a perturbative approach to a class of Fokker-Planck equations, which have constant diffusion coefficients and small time-dependent drift coefficients, was developed by exploiting the close connection between the Fokker-Planck equations and the Schrodinger equations. In this work, we further explore the possibility of similarity solutions of such a class of Fokker-Planck equations. These solutions possess definite scaling behaviors, and are obtained by means of the so-called similarity method

Green functions and nonlinear systems: Short time expansion

We show that Green function methods can be straightforwardly applied to nonlinear equations appearing as the leading order of a short time expansion. Higher order corrections can be then computed giving a satisfactory agreement with numerical results. The relevance of these results relies on the possibility of fully exploiting a gradient expansion in both classical and quantum field theory granting the existence of a strong coupling expansion. Having a Green function in this regime in quantum field theory amounts to obtain the corresponding spectrum of the theory.Comment: 7 pages, 3 figures. Version accepted for publication in International Journal of Modern Physics

Long Distance Energy Correlations in Random Media

This thesis considers the long distance motion of waves in a random medium. Using the geometrical optics approximation and a stochastic limit theorem, we find evolution equations for rays and for energy correlations, in two and three dimensions. Our equations are valid on a long distance scale, well after the focusing of rays has become significant. We construct asymptotic expansions of the two point energy correlation function in two and three dimensions. In two dimensions we numerically solve the partial differential equation that determines the two point energy correlation function. We also perform Monte-Carlo simulations to calculate the same quantity. There is good agreement between the two solutions. We present the solution for the two point energy correlation function obtained by regular perturbation techniques. This solution agrees with our solution until focusing becomes significant. Then our solution is valid (as shown by the Monte-Carlo simulations), while the regular perturbation solution becomes invalid. Also presented are the equations that describe energy correlations after a wave has gone through a weakly stochastic plane layered medium.</p

Analytical Solution of the O-X Mode Conversion Problem

The excitation of a slow extraordinary wave in a overdense plasma from an ordinary wave impinging on the critical layer in the plane spanned by the density gradient and magnetic field is solved analytically by formulating the problem in terms of a parabolic cylinder equation. A formula for the angular dependence of the transmission coefficient is derived.Comment: Final version as accepted for publication in Phys. Lett.

Game theory analysis when playing the wrong game

In classical game theory, optimal strategies are determined for games with complete information; this requires knowledge of the opponent's goals. We analyze games when a player is mistaken about their opponents goals. For definitiveness, we study the (common) bimatrix formulation where both player's payoffs are matrices. While the payoff matrix weights are arbitrary, we focus on strict ordinal payoff matrices, which can be enumerated. In this case, a reasonable error would be for one player to switch two ordinal values in their opponents payoff matrix. The mathematical formulation of this problem is stated, and all 78 strict ordinal 2-by-2 bimatrix games are investigated. This type of incomplete information game has not -- to our knowledge -- been studied before

General solutions of the Monge-Amp\`{e}re equation in nn-dimensional space

It is shown that the general solution of a homogeneous Monge-Amp\`{e}re equation in $n$-dimensional space is closely connected with the exactly (but only implicitly) integrable system \frac {\partial \xi_{j}}{\partial x_0}+\sum_{k=1}^{n-1} \xi_{k} \frac {\partial \xi_{j}}{\partial x_{k}}=0 \label{1} Using the explicit form of solution of this system it is possible to construct the general solution of the Monge-Amp\`{e}re equation.Comment: 8 page

Coarsening scenarios in unstable crystal growth

Crystal surfaces may undergo thermodynamical as well kinetic, out-of-equilibrium instabilities. We consider the case of mound and pyramid formation, a common phenomenon in crystal growth and a long-standing problem in the field of pattern formation and coarsening dynamics. We are finally able to attack the problem analytically and get rigorous results. Three dynamical scenarios are possible: perpetual coarsening, interrupted coarsening, and no coarsening. In the perpetual coarsening scenario, mound size increases in time as L=t^n, where the coasening exponent is n=1/3 when faceting occurs, otherwise n=1/4.Comment: Changes in the final part. Accepted for publication in Phys. Rev. Let

Resolving velocity space dynamics in continuum gyrokinetics

Many plasmas of interest to the astrophysical and fusion communities are weakly collisional. In such plasmas, small scales can develop in the distribution of particle velocities, potentially affecting observable quantities such as turbulent fluxes. Consequently, it is necessary to monitor velocity space resolution in gyrokinetic simulations. In this paper, we present a set of computationally efficient diagnostics for measuring velocity space resolution in gyrokinetic simulations and apply them to a range of plasma physics phenomena using the continuum gyrokinetic code GS2. For the cases considered here, it is found that the use of a collisionality at or below experimental values allows for the resolution of plasma dynamics with relatively few velocity space grid points. Additionally, we describe implementation of an adaptive collision frequency which can be used to improve velocity space resolution in the collisionless regime, where results are expected to be independent of collision frequency.Comment: 20 pages, 11 figures, submitted to Phys. Plasma