Artificial intelligence (AI) has the potential to transform medical imaging by automating image analysis and accelerating clinical research. However, research and clinical use are limited by the wide variety of AI implementations and architectures, inconsistent documentation, and reproducibility issues. Here, we introduce MHub$.$ai, an open-source, container-based platform that standardizes access to AI models with minimal configuration, promoting accessibility and reproducibility in medical imaging. MHub$.$ai packages models from peer-reviewed publications into standardized containers that support direct processing of DICOM and other formats, provide a unified application interface, and embed structured metadata. Each model is accompanied by publicly available reference data that can be used to confirm model operation. MHub$.$ai includes an initial set of state-of-the-art segmentation, prediction, and feature extraction models for different modalities. The modular framework enables adaptation of any model and supports community contributions. We demonstrate the utility of the platform in a clinical use case through comparative evaluation of lung segmentation models. To further strengthen transparency and reproducibility, we publicly release the generated segmentations and evaluation metrics and provide interactive dashboards that allow readers to inspect individual cases and reproduce or extend our analysis. By simplifying model use, MHub$.$ai enables side-by-side benchmarking with identical execution commands and standardized outputs, and lowers the barrier to clinical translation.

Modern agricultural operations increasingly rely on integrated monitoring systems that combine multiple data sources for farm optimization. Aerial drone-based animal health monitoring serves as a key component but faces limited data availability, compounded by scene-specific issues such as small, occluded, or partially visible animals. Transfer learning approaches often fail to address this limitation due to the unavailability of large datasets that reflect specific farm conditions, including variations in animal breeds, environments, and behaviors. Therefore, there is a need for developing a problem-specific, animal-focused data augmentation strategy tailored to these unique challenges. To address this gap, we propose an object-focused data augmentation framework designed explicitly for animal health monitoring in constrained data settings. Our approach segments animals from backgrounds and augments them through transformations and diffusion-based synthesis to create realistic, diverse scenes that enhance animal detection and monitoring performance. Our initial experiments demonstrate that our augmented dataset yields superior performance compared to our baseline models on the animal detection task. By generating domain-specific data, our method empowers real-time animal health monitoring solutions even in data-scarce scenarios, bridging the gap between limited data and practical applicability.

We establish power saving asymptotics for the sum of the divisor function along a binary quartic form, improving on work of Daniel. The proof involves an application of a recent two dimensional delta method due to Li, Rydin-Myerson, and Vishe and an exploitation of $\mathrm{GL}_2$ automorphic forms arising from the factorization of varying cubic Dedekind zeta functions.

We discuss and draw the reader's attention to several passages in Vladimir Arnold's note on the epigraph to the novel in verse "Evgenii Onegin" by A$.$S$.$Pushkin, as well as to puzzles hidden in the novel by the poet himself.

Experiments on Bose-Einstein condensates at finite temperature typically extract the system parameters, such as temperature, atom number, and condensed fraction from time-of-flight images taken after a free expansion time. This paper systematically examines the effect of repulsive interactions between the condensed and thermal atoms in partially condensed clouds on the expansion profile of the thermal cloud. An analytical expression for the expansion can be obtained only if the interactions between the Bose-Einstein condensate and thermal atoms are neglected, resulting in a Bose-enhanced distribution for the thermal component. Here, the deformation of the cloud due to interactions and the effects on estimated parameters are investigated by simulating the expansion using a ballistic approximation. By fitting the simulated expansion profiles with a Bose-enhanced distribution, the errors of using such a fit are estimated, and the results are explained phenomenologically. The simulation was also used as a fitting function for experimental data, showing better agreement of the extracted condensed fraction with the semi-ideal model than results from a Bose-enhanced fit.

This work studies constrained blackbox optimization problems that cannot be solved in reasonable time due to prohibitive computational costs. This challenge is especially prevalent in industrial applications, where blackbox evaluations are costly. However, constraints can be evaluated at various fidelities at a lower computational cost. More specifically, this work targets situations in which the infeasibility of each individual constraint can be detected at lower fidelities, and where a large discrete number of fidelities are available. Moreover, highly discontinuous problems which may fail to evaluate are considered, such that direct search methods are preferred to model-based ones. To this effect, the Interruptible Direct Search (IDS) and the Dynamic Interruptible Direct Search (DIDS) algorithms are proposed to leverage feasibility assessments from various fidelity levels to avoid high cost evaluations. The results show highly increased performances from NOMAD when it is paired with IDS or DIDS.

We discuss representations of product systems (of $W^*$-correspondences) over the semigroup $\mathbb{Z}^n_+$ and show that, under certain pureness and Szego positivity conditions, a completely contractive representation can be dilated to an isometric representation. For $n=1,2$ this is known to hold in general (without assuming the conditions) but, for $n\geq 3$, it does not hold in general (as is known for the special case of isometric dilations of a tuple of commuting contractions). Restricting to the case of tuples of commuting contractions, our result reduces to a result of Barik, Das, Haria and Sarkar. Our dilation is explicitly constructed and we present some applications.