Trailblazer: The Legacy of Bishop Henry M. Turner During the Civil War, Reconstruction, and Jim Crowism

When the Civil War began in 1861, the conflict evoked feelings of pride, patriotism, and hatred in both blacks and whites. As the war raged on, Reverend Henry McNeal Turner ministered to his brethren serving in the United States Colored Troops (USCT), segregated units of the Union Army. Although slavery ended in 1865 with the passage of the Thirteenth Amendment, Lincoln’s plans for Reconstruction died with his assassination. The Ku Klux Klan and ex–Confederates not only regained control of the South but also resisted the federal government’s early attempts at civil rights legislation by intimidating, murdering, and disenfranchising ex–slaves. In response to the brutality and the rise of Jim Crowism, Turner served as a beacon of hope for thousands of freedmen while respectively serving in state and local politics. Bishop Henry M. Turner’s story deserves more attention because he is an overlooked transitional figure in American history. This paper will examine Turner’s contributions to the A.M.E. Church, politics, and civil rights

On the asymptotic behaviour of the correlation measure of sum-of-digits function in base 2

Let $s\_2(x)$ denote the number of digits "$1$" in a binary expansion of any $x \in \mathbb{N}$. We study the mean distribution $\mu\_a$ of the quantity $s\_2(x+a)-s\_2(x)$ for a fixed positive integer $a$.It is shown that solutions of the equation$$s\_2(x+a)-s\_2(x)= d$$are uniquely identified by a finite set of prefixes in $\{0,1\}^*$, and that the probability distribution of differences $d$ is given by an infinite product of matrices whose coefficients are operators of $l^1(\mathbb{Z})$.Then, denoting by $l(a)$ the number of patterns "$01$" in the binary expansion of $a$, we give the asymptotic behaviour of this probability distribution as $l(a)$ goes to infinity as well as estimates of the variance of the probability measure $\mu\_a

Evaluating probabilistic forecasts with scoringRules

Probabilistic forecasts in the form of probability distributions over future events have become popular in several fields including meteorology, hydrology, economics, and demography. In typical applications, many alternative statistical models and data sources can be used to produce probabilistic forecasts. Hence, evaluating and selecting among competing methods is an important task. The scoringRules package for R provides functionality for comparative evaluation of probabilistic models based on proper scoring rules, covering a wide range of situations in applied work. This paper discusses implementation and usage details, presents case studies from meteorology and economics, and points to the relevant background literature