Products of random variables and the first digit phenomenon

We provide conditions on dependent and on non-stationary random variables $X_n$ ensuring that the mantissa of the sequence of products $\left({\prod }_1^n{X}_k\right)$ is almost surely distributed following Benford’s law or converges in distribution to Benford’s law. This is achieved through proving new generalizations of Lévy’s and Robbins’s results on distribution modulo 1 of sums of independent random variables.     

A hybrid of adaptive large neighborhood search and tabu search for the order-batching problem

Given a set of customer orders and a routing policy, the goal of the order-batching problem?(OBP) is to group customer orders to picking orders (batches) such that the total length of all tours through a rectangular warehouse is minimized. Because order picking is considered the most labor-intensive process in warehousing, effectively batching customer orders can result in considerable savings. The OBP is NP-hard if the number of orders per batch is greater than two, and the exact solution methods proposed in the literature are not able to consistently solve larger instances. To address larger instances, we develop a metaheuristic hybrid based on adaptive large neighborhood search and tabu search, called ALNS/TS. In numerical studies, we conduct an extensive comparison of ALNS/TS to all previously published OBP methods that have used standard benchmark sets to investigate their performance. ALNS/TS outperforms all comparison methods with respect to both average solution quality and run-time. Compared to the state-of-the-art, ALNS/TS shows the clearest advantages on the larger instances of the existing benchmark sets, which assume a higher number of customer orders and higher capacities of the picking device. Finally, ALNS/TS is able to solve newly generated large-scale instances with up to 600 customer orders and six articles per customer order with reasonable run-times and convincing scaling behavior and robustness.     

Analysis of passive optical structures with an adaptive set of radiation modes

Passive optical structures are simulated by the method of lines using an eigenvalue routine based on the Arnoldi method, where its time efficiency is verified. A systematic approach is developed using an adaptive routine to determine selected radiation modes that must be taken into account in order to achieve a certain degree of accuracy.     

Nachweis und Bestimmung des Eisens, sowie seine Trennung von anderen Elementen

Ohne Zusammenfassung     

Ueber das Aequivalentgewicht des Wismuths

Ohne Zusammenfassung     

Ueber die Untersuchung und Werthbestimmung von Phenolen

Ohne Zusammenfassung     

Skineffekt und elektrische Leitf?higkeit in einem Argonplasma

Ohne Zusammenfassung     

Ueber die Aufbewahrung von Schwefelwasserstoffwasser

Ohne Zusammenfassung