Search

It is well known that almost every dilation of a sequence of real numbers, that diverges to $\infty$, is dense modulo~1. This paper studies the exceptional set of points -- those for which the dilation is not dense. Specifically, we consider the Hausdorff and modified box dimensions of the set of exceptional points. In particular, we show that the dimension of this set may be any number between 0 and 1. Similar results are obtained for two ``natural'' subsets of the set of exceptional points. Furthermore, the paper calculates the dimension of several sets of points, defined by certain constraints on their binary expansion.

Let $M(n, k, p)$ denote the maximum probability of the event $X_1 = X_2 = \cdots = X_n=1$ under a $k$-wise independent distribution whose marginals are Bernoulli random variables with mean $p$. A long-standing question is to calculate $M(n, k, p)$ for all values of $n,k,p$. This question has been partially addressed by several authors, primarily with the goal of answering asymptotic questions. The present paper focuses on obtaining exact expressions for this probability. To this end, we provide closed-form formulas of $M(n,k,p)$ for $p$ near 0 as well as $p$ near 1.

Two lattice points are visible from one another if there is no lattice point on the open line segment joining them. Let $S$ be a finite subset of $\mathbb{Z}^k$. The asymptotic density of the set of lattice points, visible from all points of $S$, was studied by several authors. Our main result is an improved upper bound on the error term. We also find the Schnirelmann density of the set of visible points from some sets S. Finally, we discuss these questions from the point of view of ergodic theory., Comment: Published in Journal of Number Theory

Let $(a_n)_{n=0}^\infty$ be a second-order linear recurrence sequence with constant coefficients over the field of $p$-adic numbers $\mathbb{Q}_p$. We study the set of limit points of the sequence of consecutive ratios $(a_{n+1}/a_{n})_{n=0}^\infty$ in $\mathbb{Q}_p$., Comment: to appear in the International Journal of Number Theory

The recent advent of autonomous laboratories, coupled with algorithms for high-throughput screening and active learning, promises to accelerate materials discovery and innovation. As these autonomous systems grow in complexity, the demand for robust and efficient workflow management software becomes increasingly critical. In this paper, we introduce AlabOS, a general-purpose software framework for orchestrating experiments and managing resources, with an emphasis on automated laboratories for materials synthesis and characterization. AlabOS features a reconfigurable experiment workflow model and a resource reservation mechanism, enabling the simultaneous execution of varied workflows composed of modular tasks while eliminating conflicts between tasks. To showcase its capability, we demonstrate the implementation of AlabOS in a prototype autonomous materials laboratory, A-Lab, with around 3,500 samples synthesized over 1.5 years., Comment: 34 pages, 5 figures

Adding a large A-site cation, such as dimethylammonium (DMA), to the perovskite growth solution has been shown to improve performance and long-term operational stability of halide perovskite solar cells. To better understand the origins of these improvements, we explore the changes in film structure, composition, and optical properties of a formamidinium (FA), Cs, Pb, mixed halide perovskite following addition of DMA to the perovskite growth solution in the ratio DMA0.1FA0.6Cs0.3Pb(I0.8Br0.2)3. Using time-of-flight secondary-ion mass spectrometry (TOF-SIMS) we show that DMA is indeed incorporated into the perovskite, with a higher DMA concentration at the surface. Using a combination of PL microscopy and photo-induced force microscopy-based (PiFM) nanoinfrared (nanoIR), we demonstrate that incorporating DMA into the film leads to increased local heterogeneity in the local bandgap, and clustering of the local formamidinium (CH5N2+) composition. In addition, using nano-X-ray diffraction, we demonstrate that DMA incorporation also alters the local structural composition by changing the local d-spacing distribution and grain size. Our results suggest that compositional variations in the organic cations at the A-site drive the structural heterogeneity observed in case of DMA incorporated films. Our results also suggest that while current-DMA-additive based approaches do have benefits to operational stability and device performance, process optimization to achieve local compositional and structural homogeneity could further boost both of these gains in performance, bringing further gains to solar cells using DMA additives.

While halide perovskites attract significant academic attention, examples of at-scale industrial production are still sparse. In this perspective, we review practical challenges hindering the commercialization of halide perovskites, and discuss how machine-learning (ML) tools could help: (1) active-learning algorithms that blend institutional knowledge and human expertise could help stabilize and rapidly update baseline manufacturing processes; (2) ML-powered metrology, including computer imaging, could help narrow the performance gap between large- and small-area devices; and (3) inference methods could help accelerate root-cause analysis by reconciling multiple data streams and simulations, focusing research effort on areas with highest probability for improvement. We conclude that to satisfy many of these challenges, incremental -- not radical -- adaptations of existing ML and statistical methods are needed. We identify resources to help develop in-house data-science talent, and propose how industry-academic partnerships could help adapt "ready-now" ML tools to specific industry needs, further improve process control by revealing underlying mechanisms, and develop "gamechanger" discovery-oriented algorithms to better navigate vast materials combination spaces and the literature., Comment: 21 pages, 2 figures