Symmetry, Entropy, Diversity and (why not?) Quantum Statistics in Society

We describe society as a nonequilibrium probabilistic system: N individuals occupy W resource states in it and produce entropy S over definite time periods. Resulting thermodynamics is however unusual because a second entropy, H, measures a typically social feature, inequality or diversity in the distribution of available resources. A symmetry phase transition takes place at Gini values 1/3, where realistic distributions become asymmetric. Four constraints act on S: expectedly, N and W, and new ones, diversity and interactions between individuals; the latter result from the two coordinates of a single point in the data, the peak. The occupation number of a job is either zero or one, suggesting Fermi-Dirac statistics for employment. Contrariwise, an indefinite nujmber of individuals can occupy a state defined as a quantile of income or of age, so Bose-Einstein statistics may be required. Indistinguishability rather than anonymity of individuals and resources is thus needed. Interactions between individuals define define classes of equivalence that happen to coincide with acceptable definitions of social classes or periods in human life. The entropy S is non-extensive and obtainable from data. Theoretical laws are compared to data in four different cases of economical or physiological diversity. Acceptable fits are found for all of them.Comment: 13 pages, 2 figure

The Aftereffects of TC Heartland: How to Effectively Approach Motions to Dismiss and Motions to Transfer on the Basis of Improper Venue

Prior to the Supreme Court\u27s decision in TC Heartland, the law of venue in patent infringement actions fluctuated over time. In recent history, the Eastern District of Texas became a notoriously plaintiff-friendly forum in which to litigate patent infringement actions; it was also a widely available choice of forum due to the Court of Appeals for the Federal Circuit\u27s broad reading of the patent venue statute, 28 U.S.C. § 1400(b). However, the Supreme Court in TC Heartland adopted its earlier interpretation of the patent venue statute that is much narrower than subsequent interpretive expansions. This Note surveys and categorizes motions to dismiss and motions to transfer on the basis of improper venue in patent infringement actions in the post-TC Heartland era through an overview of applicable law and an analysis of motion outcomes. The Note concludes with an issue-specific explanation of trends in such motion outcomes, suggests that the Court of Appeals for the Federal Circuit\u27s recent decision to place the burden of proof in these motions on plaintiffs will result in disproportionate victories for defendants, and proposes strategies for plaintiffs to mitigate this burden

On the Optimality of Averaging in Distributed Statistical Learning

A common approach to statistical learning with big-data is to randomly split it among $m$ machines and learn the parameter of interest by averaging the $m$ individual estimates. In this paper, focusing on empirical risk minimization, or equivalently M-estimation, we study the statistical error incurred by this strategy. We consider two large-sample settings: First, a classical setting where the number of parameters $p$ is fixed, and the number of samples per machine $n\to\infty$. Second, a high-dimensional regime where both $p,n\to\infty$ with $p/n \to \kappa \in (0,1)$. For both regimes and under suitable assumptions, we present asymptotically exact expressions for this estimation error. In the fixed-$p$ setting, under suitable assumptions, we prove that to leading order averaging is as accurate as the centralized solution. We also derive the second order error terms, and show that these can be non-negligible, notably for non-linear models. The high-dimensional setting, in contrast, exhibits a qualitatively different behavior: data splitting incurs a first-order accuracy loss, which to leading order increases linearly with the number of machines. The dependence of our error approximations on the number of machines traces an interesting accuracy-complexity tradeoff, allowing the practitioner an informed choice on the number of machines to deploy. Finally, we confirm our theoretical analysis with several simulations.Comment: Major changes from previous version. Particularly on the second order error approximation and implication

Geometric and Measure-Theoretic Shrinking Targets in Dynamical Systems

We consider both geometric and measure-theoretic shrinking targets for ergodic maps, investigating when they are visible or invisible. Some Baire category theorems are proved, and particular constructions are given when the underlying map is fixed. Open questions about shrinking targets are also described

Estimation for almost periodic processes

Processes with almost periodic covariance functions have spectral mass on lines parallel to the diagonal in the two-dimensional spectral plane. Methods have been given for estimation of spectral mass on the lines of spectral concentration if the locations of the lines are known. Here methods for estimating the intercepts of the lines of spectral concentration in the Gaussian case are given under appropriate conditions. The methods determine rates of convergence sufficiently fast as the sample size $n\to\infty$ so that the spectral estimation on the estimated lines can then proceed effectively. This task involves bounding the maximum of an interesting class of non-Gaussian possibly nonstationary processes.Comment: Published at http://dx.doi.org/10.1214/009053606000000218 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org