For the deterministic dyadic model of turbulence, there are examples of initial conditions in $l^{2}$ which have more than one solution. The aim of this paper is to prove that uniqueness, for all $l^{2}$ -initial conditions, is restored when a suitable multiplicative noise is introduced. The noise is formally energy preserving. Uniqueness is understood in the weak probabilistic sense.

Estimating the co-expression of cell identity factors in single-cell is crucial. Due to the low efficiency of scRNA-seq methodologies, sensitive computational approaches are critical to accurately infer transcription profiles in a cell population. We introduce COTAN, a statistical and computational method, to analyze the co-expression of gene pairs at single cell level, providing the foundation for single-cell gene interactome analysis. The basic idea is studying the zero UMI counts’ distribution instead of focusing on positive counts; this is done with a generalized contingency tables framework. COTAN can assess the correlated or anti-correlated expression of gene pairs, providing a new correlation index with an approximate p-value for the associated test of independence. COTAN can evaluate whether single genes are differentially expressed, scoring them with a newly defined global differentiation index. Similarly to correlation network analysis, it provides ways to plot and cluster genes according to their co-expression pattern with other genes, effectively helping the study of gene interactions, becoming a new tool to identify cell-identity markers. We assayed COTAN on two neural development datasets with very promising results. COTAN is an R package that complements the traditional single cell RNA-seq analysis and it is available at https://github.com/seriph78/COTAN.

Coevolution at the gene level, as reflected by correlated events of gene loss or gain, can be revealed by phylogenetic profile analysis. The optimal method and metric for comparing phylogenetic profiles, especially in eukaryotic genomes, are not yet established. Here, we describe a procedure suitable for large-scale analysis, which can reveal coevolution based on the assessment of the statistical significance of correlated presence/absence transitions between gene pairs. This metric can identify coevolution in profiles with low overall similarities and is not affected by similarities lacking coevolutionary information. We applied the procedure to a large collection of 60,912 orthologous gene groups (orthogroups) in 1,264 eukaryotic genomes extracted from OrthoDB. We found significant cotransition scores for 7,825 orthogroups associated in 2,401 coevolving modules linking known and unknown genes in protein complexes and biological pathways. To demonstrate the ability of the method to predict hidden gene associations, we validated through experiments the involvement of vertebrate malate synthase-like genes in the conversion of ( S )-ureidoglycolate into glyoxylate and urea, the last step of purine catabolism. This identification explains the presence of glyoxylate cycle genes in metazoa and suggests an anaplerotic role of purine degradation in early eukaryotes.

Managing blood glucose levels in Type 1 Diabetes Mellitus (T1DM) is essential to prevent complications. Traditional insulin delivery methods often require significant patient involvement, limiting automation. Reinforcement Learning (RL)-based controllers offer a promising approach for improving automated insulin administration. We propose a Dual Proximal Policy Optimization (Dual PPO) controller for personalized insulin delivery in a hybrid closed-loop system. The controller optimizes patient-specific insulin bounds through a grid search on pre-trained models to manage both hyperglycemia and hypoglycemia. A safe-control mechanism prevents insulin administration when glucose levels drop below a predefined threshold. The system was evaluated on 10 in silico adult patients using the UVA/Padova simulator, with five-day randomized meal scenarios. The Dual PPO controller significantly improved Time in Range (TIR) (69.30% ±1.61) compared to a single PPO model (61.69% ±1.54). The system effectively reduced severe hyperglycemia while maintaining a low incidence of severe hypoglycemia. Unlike conventional open-loop methods such as Basal-Bolus (BBC) and Proportional-Integral-Derivative (PIDC) controllers, our system requires minimal patient interaction, eliminating the need for carbohydrate estimation. The Dual PPO controller enhances personalized insulin delivery in T1DM, improving glycemic control while reducing patient burden. This approach advances precision medicine in diabetes management, with potential for future real-world applications.