Mal'cev nilpotent algebras

We show that the Mal'cev semigroup identity x_n = y_n holds in the circle semigroup of an associative algebra over an infinite field precisely when the algebra is Lie nilpotent of class at most n. The Mal'cev semigroup law x_n = y_n holds in a group if and only if the group is nilpotent of class at most n.     

Golem: an algorithm for robust experiment and process optimization

Numerous challenges in science and engineering can be framed as optimization tasks, including the maximization of reaction yields, the optimization of molecular and materials properties, and the fine-tuning of automated hardware protocols. Design of experiment and optimization algorithms are often adopted to solve these tasks efficiently. Increasingly, these experiment planning strategies are coupled with automated hardware to enable autonomous experimental platforms. The vast majority of the strategies used, however, do not consider robustness against the variability of experiment and process conditions. In fact, it is generally assumed that these parameters are exact and reproducible. Yet some experiments may have considerable noise associated with some of their conditions, and process parameters optimized under precise control may be applied in the future under variable operating conditions. In either scenario, the optimal solutions found might not be robust against input variability, affecting the reproducibility of results and returning suboptimal performance in practice. Here, we introduce Golem, an algorithm that is agnostic to the choice of experiment planning strategy and that enables robust experiment and process optimization. Golem identifies optimal solutions that are robust to input uncertainty, thus ensuring the reproducible performance of optimized experimental protocols and processes. It can be used to analyze the robustness of past experiments, or to guide experiment planning algorithms toward robust solutions on the fly. We assess the performance and domain of applicability of Golem through extensive benchmark studies and demonstrate its practical relevance by optimizing an analytical chemistry protocol under the presence of significant noise in its experimental conditions.

Numerous challenges in science and engineering can be framed as optimization tasks. Golem is an uncertain-input algorithm that ensures the reproducible performance of optimized experimental protocols and processes.