This paper investigates the optimal control problem for a class of parabolic equations where the diffusion coefficient is influenced by a control function acting nonlocally. Specifically, we consider the optimization of a cost functional that incorporates a controlled probability density evolving under a Fokker-Planck equation with state-dependent drift and diffusion terms. The control variable is subject to spatial convolution through a kernel, inducing nonlocal interactions in both drift and diffusion terms. We establish the existence of optimal controls under appropriate convexity and regularity conditions, leveraging compactness arguments in function spaces. A maximum principle is derived to characterize the optimal control explicitly, revealing its dependence on the adjoint state and the nonlocal structure of the system. We further provide a rigorous financial application in the context of mean-variance portfolio optimization, where both the asset drift and volatility are controlled nonlocally, leading to an integral representation of the optimal investment strategy. The results offer a mathematically rigorous framework for optimizing diffusion-driven systems with spatially distributed control effects, broadening the applicability of nonlocal control methods to stochastic optimization and financial engineering.
This work investigates a novel approach for the high order evolution of hyperbolic PDEs using ADER discontinuous Galerkin schemes within a direct Arbitrary-Lagrangian-Eulerian (ALE) framework on 3D moving polyhedral meshes with topology changes. Our direct ALE method is based on the PDE integration over 4D (3D+time) space-time control volumes connecting the elements of two subsequent tessellations, so to simultaneously evolve the solution both in time and between the two different meshes in an effective and high order manner. In this way, we also avoid any complex and expensive projection-reconstruction techniques and any mesh intersection operation typical of indirect ALE schemes. The crucial step consists in the strategy for building space-time control volumes that also connect elements with different shapes and neighborhoods due to a change in topology. In fact, simply linking existing elements by collapsing or expanding their edges would leave a "hole" in the space-time domain. To fill it, we introduce additional degenerate elements that we call hole-like elements. These are 4D objects with zero 3D volume at both the beginning and end of the timestep, but which possess a strictly non-zero 4D space-time volume. Given the uniqueness of this space-time approach in 3D+time and the necessity of characterizing the geometry of such elements, the main objective of this paper is the formal geometrical and numerical description of the method. Specifically, we show that despite being non-trivial to visualize, these elements are well-defined and ultimately even easy to manage. In particular, we provide here a detailed characterization of the hole-like elements needed to connect two polyhedral tessellations, corresponding to 2-3, 3-2, and 4-4 flips on the underlying Delaunay tetrahedralization. We describe how to identify and construct them, how to adapt the numerical method to their degenerate geometry by introducing for them a locally implicit treatment, and we provide new and intuitive visualization strategies. Finally, a set of benchmarks, simple in their wave structure but challenging for moving meshes, is used to prove that the method is fully conservative, satisfies the Geometric Conservation Law condition, and maintains the correct order of convergence even in the presence of frequent topology changes.
Background Functional Motor Disorders (FMDs) are increasingly recognized in pediatrics, yet prognostic indicators remain poorly characterized. We examined clinical and health-care variables associated with one-year clinical outcomes. Methods This retrospective study included 72 patients aged 9-18 years diagnosed with FMD at two tertiary pediatric hospitals in Italy and the Czech Republic between 2019 and 2023, with at least 6 months of follow-up and two valid Simplified Functional Movement Disorders Rating Scale assessments. Demographic and clinical features, referral history, therapy type and timing, and Simplified Functional Movement Disorders Rating Scale scores were analyzed at baseline, 6, and 12 months. Results Overall, 69.4% were female, and mean symptom-onset age was 13.5 ± 1.99 years. Motor severity improved from baseline (10.25) to 12 months (3.55; P < 0.001), with 43.1% achieving remission. Longer symptom duration was independently associated with higher severity at baseline (β = 1.19, P = 0.021), 6 months (β = 2.07, P < 0.001), and 12 months (β = 1.79, P < 0.001). More prior referrals independently predicted higher 12-month severity (β = 1.00, P = 0.047). Longer time to FMD-directed treatment was associated with higher follow-up severity and a different symptom trajectory (Time × treatment timing: P = 0.007). Trajectories did not consistently differ by sex or symptom phenotype. Conclusions Longer symptom duration, more previous referrals, and delayed treatment were associated with greater one-year motor severity. These findings underscore the prognostic relevance of symptom duration and the potential importance of timely access to care, although prospective studies are needed to establish causality.
Long-term animal re-identification (ReID) must remain robust to gradual morphological evolution and seasonal appearance shifts. Although recent vision-language models provide strong pretrained visual representations, adapting them to longitudinal ecological settings remains challenging, particularly under identity and temporal distribution shifts. We present a parameter-efficient CLIP adaptation framework for animal ReID and introduce a continuous metadata-conditioning mechanism that incorporates numerical attributes directly into the prompt representation during training. While low-rank visual adaptation, prompt-based supervision, and cross-modal alignment provide the adaptation framework, the proposed metadata-conditioning strategy constitutes the primary methodological contribution. By preserving the continuous structure of numerical metadata rather than discretizing it into textual categories, the proposed approach enables smooth modulation of the embedding space during training while maintaining a purely visual inference pipeline. Experiments on a seven-year longitudinal fish dataset and multiple wildlife benchmarks demonstrate improved performance under closed-set, open-set, and time-aware evaluation protocols. The results demonstrate that continuous metadata conditioning improves robustness to longitudinal appearance variation and temporal distribution shifts, while parameter-efficient adaptation enables a purely visual inference pipeline without requiring metadata at test time. Code and evaluation splits can be found at: https://github.com/AnilOsmanTur/MetaPrompt-ReID.
