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During the Second World War, estimates of the number of tanks deployed by Germany were critically needed. The Allies adopted a successful statistical approach to estimate this information: assuming that the tanks are sequentially numbered starting from 1, if we observe k tanks from an unknown total of N, then the best linear unbiased estimator for N is M(1+1/k)-1 where M is the maximum observed serial number. However, in many situations, the original German Tank Problem is insufficient, since typically there are l > 1 factories, and tanks produced by different factories may have serial numbers in disjoint ranges that are often far separated.Clark, Gonye and Miller presented an unbiased estimator for N when the minimum serial number is unknown. Provided one identifies which samples correspond to which factory, one can then estimate each factory's range and summing the sizes of these ranges yields an estimate for the rival's total productivity. We construct an efficient procedure to estimate the total productivity and prove that it is effective when log l/log k is sufficiently small. In the final section, we show that given information about the gaps, we can make an estimator that performs orders of magnitude better when we have a small number of samples. 

In 1946, Erd\H{o}s posed the distinct distance problem, which seeks to findthe minimum number of distinct distances between pairs of points selected fromany configuration of $$n$$ points in the plane. The problem has since beenexplored along with many variants, including ones that extend it into higherdimensions. Less studied but no less intriguing is Erd\H{o}s' distinct angleproblem, which seeks to find point configurations in the plane that minimizethe number of distinct angles. In their recent paper "Distinct Angles inGeneral Position," Fleischmann, Konyagin, Miller, Palsson, Pesikoff, and Wolfuse a logarithmic spiral to establish an upper bound of $$O(n^2)$ on the minimumnumber of distinct angles in the plane in general position, which prohibitsthree points on any line or four on any circle. We consider the question of distinct angles in three dimensions and providebounds on the minimum number of distinct angles in general position in thissetting. We focus on pinned variants of the question, and we examine explicitconstructions of point configurations in $$\mathbb{R}^3$$ which useself-similarity to minimize the number of distinct angles. Furthermore, westudy a variant of the distinct angles question regarding distinct angle chainsand provide bounds on the minimum number of distinct chains in $$\mathbb{R}^2$$and $$\mathbb{R}^3$$. 

Abstract The detection of multilayer clouds in the atmosphere can be particularly challenging from passive visible and infrared imaging radiometers since cloud boundary information is limited primarily to the topmost cloud layer. Yet detection of low clouds in the atmosphere is important for a number of applications, including aviation nowcasting and general weather forecasting. In this work, we develop pixel-based machine learning–based methods of detecting low clouds, with a focus on improving detection in multilayer cloud situations and specific attention given to improving the Cloud Cover Layers (CCL) product, which assigns cloudiness in a scene into vertical bins. The random forest (RF) and neural network (NN) implementations use inputs from a variety of sources, including GOES Advanced Baseline Imager (ABI) visible radiances, infrared brightness temperatures, auxiliary information about the underlying surface, and relative humidity (which holds some utility as a cloud proxy). Training and independent validation enlists near-global, actively sensed cloud boundaries from the radar and lidar systems on board theCloudSatandCALIPSOsatellites. We find that the RF and NN models have similar performances. The probability of detection (PoD) of low cloud increases from 0.685 to 0.815 when using the RF technique instead of the CCL methodology, while the false alarm ratio decreases. The improved PoD of low cloud is particularly notable for scenes that appear to be cirrus from an ABI perspective, increasing from 0.183 to 0.686. Various extensions of the model are discussed, including a nighttime-only algorithm and expansion to other satellite sensors. Significance StatementUsing satellites to detect the heights of clouds in the atmosphere is important for a variety of weather applications, including aviation weather forecasting. However, detecting low clouds can be challenging if there are other clouds above them. To address this, we have developed machine learning–based models that can be used with passive satellite instruments. These models use satellite observations at visible and infrared wavelengths, an estimate of relative humidity in the atmosphere, and geographic and surface-type information to predict whether low clouds are present. Our results show that these models have significant skill at predicting low clouds, even in the presence of higher cloud layers. 

Genetic screens are valuable for identifying novel genes involved in the regulation of developmental processes. To identify genes associated with cell growth regulation in Drosophila melanogaster, a mutagenesis screen was performed. Undergraduate students participating in Fly-CURE phenotypically characterized the E.4.1 mutant which is associated with rough eyes and antennae overgrowth. Following complementation analysis and subsequent genomic sequencing, E.4.1 was identified as a novel mutant allele of GstE14, a gene involved in ecdysone biosynthesis important for the timing of developmental events. The abnormal eye and antenna phenotypes observed resulting from the loss of GstE14 suggest its role in tissue growth.