Mathematics

Accurate segmentation of the pancreas in Computed Tomography (CT) scans is a challenging task, which may be crucial for the diagnosis and treatment planning of pancreatic cancer. The irregular shape of the pancreas, its low contrast relative to surrounding tissues, and its close proximity to other complex anatomical structures make it difficult to accurately delineate its contours. Furthermore, a significant class imbalance between foreground (pancreas) and background voxels in an abdominal CT series represents an additional challenge for deep-learning-based approaches. In this study, we developed a deep learning model for automated pancreas segmentation based on a 3D U-Net architecture enhanced with an attention mechanism to improve the model capability to focus on relevant anatomical features of the pancreas. The model was trained and evaluated on two widely used benchmark datasets for volumetric segmentation, the NIH Healthy Pancreas-dataset and the Medical Segmentation Decathlon (MSD) pancreas dataset. The proposed attention-guided 3D U-Net achieved a Dice score of 80.8 ± 2.1%, ASSD of 2.1 ± 0.3 mm, and HD₉₅ of 8.1 ± 1.6 mm on the NIH dataset, and the values of 78.1 ± 1.1%, 3.3 ± 0.3 mm, and 12.3 ± 1.5 mm for the same metrics on the MSD dataset, demonstrating the value of attention mechanisms in improving segmentation performance in complex and low-contrast anatomical regions.

In this article, we develop a novel kernel-based estimation framework for functional regression models in the presence of missing responses, with particular emphasis on the Missing At Random (MAR) mechanism. The analysis is carried out in the setting of stationary and ergodic functional data, where we introduce apparently for the first time a local linear estimator of the regression operator. The principal theoretical contributions of the paper may be summarized as follows. First, we establish almost sure uniform rates of convergence for the proposed estimator, thereby quantifying its asymptotic accuracy in a strong sense. Second, we prove its asymptotic normality, which provides the foundation for distributional approximations and subsequent inference. Third, we derive explicit closed-form expressions for the associated asymptotic variance, yielding a precise characterization of the limiting law. These results are obtained under standard structural assumptions on the relevant functional classes and under mild regularity conditions on the underlying model, ensuring broad applicability of the theory. On the methodological side, the asymptotic analysis is exploited to construct pointwise confidence regions for the regression operator, thereby enabling valid statistical inference. Furthermore, a comprehensive set of simulation experiments is conducted, demonstrating that the proposed estimator exhibits superior finite-sample predictive performance when compared to existing procedures, while simultaneously retaining robustness in the presence of missingness governed by MAR mechanisms.

The rapid growth of Waste Electrical and Electronic Equipment (WEEE) in the European Union highlights the need for a rigorous understanding of its long-term dynamics and the role of innovation in shaping its trajectory. This study investigates how innovation influences the dynamics of WEEE generation in the European Union. We develop an innovation-adjusted mathematical model of e-waste as a stock flow system and prove the existence and global stability of a unique positive equilibrium. The model analytically generates an environmental Kuznets-type turning point and shows that innovation reduces waste accumulation by accelerating effective depreciation. To link the theoretical results with empirical patterns, we embed the model in a STIRPAT panel specification using annual data for 27 EU member states from 2013 to 2023, where EU Eco-innovation Index (EEI) serves as a composite index which directly captures policy-driven green technology and circular economy activities, aligning precisely with our theoretical framework. We also extend the quasi-demeaning transformation to panels with correlated shocks and establish its consistency under a factor structured error process. The empirical estimates confirm a positive effect of income on WEEE at lower development levels and a negative coefficient on its squared term, consistent with an inverted U pattern, while innovation is associated with lower waste intensity. These findings demonstrate how mathematical modeling can strengthen the interpretation of macro panel evidence on circularity and provide a basis for future optimization of innovation driven sustainability transitions.