Equilibrium states of the pressure function for products of matrices

Let $\{M_i\}_{i=1}^\ell$ be a non-trivial family of $d\times d$ complex matrices, in the sense that for any $n\in \N$, there exists $i_1... i_n\in \{1,..., \ell\}^n$ such that $M_{i_1}... M_{i_n}\neq {\bf 0}$. Let $P \colon (0,\infty)\to \R$ be the pressure function of $\{M_i\}_{i=1}^\ell$. We show that for each $q>0$, there are at most $d$ ergodic $q$-equilibrium states of $P$, and each of them satisfies certain Gibbs property.Comment: 12 pages. To appear in DCD

On Binscatter

Binscatter is very popular in applied microeconomics. It provides a flexible, yet parsimonious way of visualizing and summarizing large data sets in regression settings, and it is often used for informal evaluation of substantive hypotheses such as linearity or monotonicity of the regression function. This paper presents a foundational, thorough analysis of binscatter: we give an array of theoretical and practical results that aid both in understanding current practices (i.e., their validity or lack thereof) and in offering theory-based guidance for future applications. Our main results include principled number of bins selection, confidence intervals and bands, hypothesis tests for parametric and shape restrictions of the regression function, and several other new methods, applicable to canonical binscatter as well as higher-order polynomial, covariate-adjusted and smoothness-restricted extensions thereof. In particular, we highlight important methodological problems related to covariate adjustment methods used in current practice. We also discuss extensions to clustered data. Our results are illustrated with simulated and real data throughout. Companion general-purpose software packages for \texttt{Stata} and \texttt{R} are provided. Finally, from a technical perspective, new theoretical results for partitioning-based series estimation are obtained that may be of independent interest

Time-dependent occurrence rate of electromagnetic cyclotron waves in the solar wind: evidence for effect of alpha particles?

Previous studies revealed that electromagnetic cyclotron waves (ECWs) near the proton cyclotron frequency exist widely in the solar wind, and the majority of ECWs are left-handed (LH) polarized waves. Using the magnetic field data from the STEREO mission, this Letter carries out a survey of ECWs over a long period of 7 years, and calculates the occurrence rates of ECWs with different polarization senses. Results show that the occurrence rate is nearly a constant for the ECWs with right-handed polarization, but it varies significantly for the ECWs with LH polarization. Further investigation of plasma conditions reveals that the LH ECWs take place preferentially in a plasma characterized by higher temperature, lower density, and larger velocity. Some considerable correlations between the occurrence rate of LH ECWs and the properties of ambient plasmas are discussed. The present research may provide evidence for effect of alpha particles on generation of ECWs.Comment: 17 pages, 4 figures, accepted for publication in ApJ