Generalized Schwarzschild's method

We describe a new finite element method (FEM) to construct continuous equilibrium distribution functions of stellar systems. The method is a generalization of Schwarzschild's orbit superposition method from the space of discrete functions to continuous ones. In contrast to Schwarzschild's method, FEM produces a continuous distribution function (DF) and satisfies the intra element continuity and Jeans equations. The method employs two finite-element meshes, one in configuration space and one in action space. The DF is represented by its values at the nodes of the action-space mesh and by interpolating functions inside the elements. The Galerkin projection of all equations that involve the DF leads to a linear system of equations, which can be solved for the nodal values of the DF using linear or quadratic programming, or other optimization methods. We illustrate the superior performance of FEM by constructing ergodic and anisotropic equilibrium DFs for spherical stellar systems (Hernquist models). We also show that explicitly constraining the DF by the Jeans equations leads to smoother and/or more accurate solutions with both Schwarzschild's method and FEM.Comment: 14 pages, 7 Figures, Submitted to MNRA

A Hydraulic Approach to Equilibria of Resource Selection Games

Drawing intuition from a (physical) hydraulic system, we present a novel framework, constructively showing the existence of a strong Nash equilibrium in resource selection games (i.e., asymmetric singleton congestion games) with nonatomic players, the coincidence of strong equilibria and Nash equilibria in such games, and the uniqueness of the cost of each given resource across all Nash equilibria. Our proofs allow for explicit calculation of Nash equilibrium and for explicit and direct calculation of the resulting (unique) costs of resources, and do not hinge on any fixed-point theorem, on the Minimax theorem or any equivalent result, on linear programming, or on the existence of a potential (though our analysis does provide powerful insights into the potential, via a natural concrete physical interpretation). A generalization of resource selection games, called resource selection games with I.D.-dependent weighting, is defined, and the results are extended to this family, showing the existence of strong equilibria, and showing that while resource costs are no longer unique across Nash equilibria in games of this family, they are nonetheless unique across all strong Nash equilibria, drawing a novel fundamental connection between group deviation and I.D.-congestion. A natural application of the resulting machinery to a large class of constraint-satisfaction problems is also described.Comment: Hebrew University of Jerusalem Center for the Study of Rationality discussion paper 67

An Alternative Approach to Functional Linear Partial Quantile Regression

We have previously proposed the partial quantile regression (PQR) prediction procedure for functional linear model by using partial quantile covariance techniques and developed the simple partial quantile regression (SIMPQR) algorithm to efficiently extract PQR basis for estimating functional coefficients. However, although the PQR approach is considered as an attractive alternative to projections onto the principal component basis, there are certain limitations to uncovering the corresponding asymptotic properties mainly because of its iterative nature and the non-differentiability of the quantile loss function. In this article, we propose and implement an alternative formulation of partial quantile regression (APQR) for functional linear model by using block relaxation method and finite smoothing techniques. The proposed reformulation leads to insightful results and motivates new theory, demonstrating consistency and establishing convergence rates by applying advanced techniques from empirical process theory. Two simulations and two real data from ADHD-200 sample and ADNI are investigated to show the superiority of our proposed methods