In 2008, Halman showed that for any finite set P⊂Rd and any finite family B of axis-parallel boxes in Rd, if the intersection of P and any subfamily B′⊆B of size at most 2d is non-empty, then the intersection of P and B is also non-empty. Very recently Edwards and Soberón initiated the study of quantitative colorful version for 2d families, (p,q)-type variation for p≥q≥d+1, and other extensions of this Helly-type result by Halman.In this paper, we study the quantitative colorful Halman problem for 2d−1 families as well its (p,q)-type variation for p≥q≥2. Specifically, our main result asserts that for any finite set P and finite families of boxes B1,…,B2d−1 in Rd, where d≥2, if every transversal B for the families has an intersection ⋂B containing at least n points of P, then there exist j∈[2d−1] and a subset of P of size at most2n+⌊n−1d⋅2d−1⌋, such that each box of Bj contains at least n points of this subset.
In blockchain networks, the strategic ordering of transactions within blocks has emerged as a significant source of profit extraction, known as Maximal Extractable Value (MEV). The transition from spam-based Priority Gas Auctions (PGA) to structured auction mechanisms like Polygon Atlas has transformed MEV extraction from public bidding wars into sealed-bid competitions under extreme time constraints. While this shift reduces network congestion, it introduces complex strategic challenges where searchers must make optimal bidding decisions within a sub-second window without knowledge of competitor behavior or presence. Traditional equilibrium-based game-theoretic models struggle in this high-frequency, partially observable environment. While auction theory provides equilibrium solutions for sealed-bid formats under incomplete information, these models typically assume known bidder value distributions and stationary competition-assumptions that are difficult to satisfy in dynamic, sub-second auctions where competitor presence and strategies evolve rapidly. We present a reinforcement learning framework for MEV extraction on Polygon Atlas and make three contributions: (1) A novel simulation environment that accurately models the stochastic arrival of arbitrage opportunities and probabilistic competition in Atlas auctions; (2) A PPO-based bidding agent optimized for real-time constraints, capable of adaptive strategy formulation in continuous action spaces while maintaining production-ready inference speeds; (3) Empirical validation demonstrating our history-conditioned agent achieves 49% Maximum-Profit Capture when deployed alongside existing searchers and a 43% relative profit improvement over the historical market leader in counterfactual replacement, significantly outperforming static bidding strategies. Our work establishes that reinforcement learning provides a critical advantage in high-frequency MEV environments where traditional optimization methods fail, offering immediate value for industrial participants and protocol designers alike.
Accurate prediction of complex formulation viscosity is crucial for the lubricant industry and engineering processes. However, developing generalizable models remains challenging due to the complexity of industrial mixtures and the scarcity of public experimental data, often resulting from corporate proprietary restrictions. This study presents a machine-learning workflow for viscosity estimation from a proprietary pseudonymized dataset, showing that robust models can be trained under data-protection limitations. We evaluated multiple machine learning architectures, including Decision Tree, Random Forest, Support Vector Regression, and Gradient Boosting, employing feature selection strategies adapted to pseudonymized data. Additionally, the Walther formula was used as a physics-based reference. While this formula required deanonymized multi-temperature viscosity data, it proved valuable for identifying experimental outliers. Among the evaluated models, Gradient Boosting demonstrated superior performance, achieving a median absolute percentage error of 16.2% on the held-out test set. These results indicate that machine learning can support viscosity screening for complex industrial fluids without requiring full compositional transparency.
In the present paper we are dealing with reflection equation algebras (R) pound corresponding to even skew-invertible Hecke symmetries. Our main result consists in computing the characters of the spectral values of the generating matrix L of (R) pound in finite-dimensional representations labeled by partitions of integers. As is known, the spectral values belong to an algebraic extension of the center of the reflection equation algebra and elements of the center can be presented as symmetric functions in spectral values. As an application of our approach, we calculate the characters of the power sums TrR(Ln) in the mentioned finite dimensional representations. In a particular case of the Drinfeld-Jimbo R-matrix the enveloping algebra U(gl(N)) can be obtained as a specific limit of the reflection equation algebra. In this limit our results for power sums coincide with those obtained in [12]. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
Rational modification and enhancement of protein function are extremely important both for understanding the fundamental principles of operation of natural macromolecules and for practical applications. Recent flourishing of machine learning-based methods in structural biology led to noteworthy advances in protein structure prediction tasks, which in turn enabled development of versatile protein engineering approaches. Among them, the methods aimed at predicting the sequence that would fold into a conditioned structure proved particularly useful. Here, we review the diversity of these methods, their applications in basic and applied research, and discuss the challenges that remain to be overcome in the future.
