We study, for 1 ≤ p ≤ ∞ , the Hardy space h e p ( B ) , the elastic analogue of the classical Hardy spaces of harmonic functions in the unit ball of R 3 . The space consists of vector-field solutions of the Lamé system satisfying the standard integrability condition on concentric spheres centered at the origin. Using the elastic Poisson kernel, we establish a Fatou-type theorem and show that h e p ( B ) is isomorphic to the R 3 -valued Lebesgue space L p on the unit sphere for 1 < p ≤ ∞ , while h e 1 ( B ) corresponds to the space of R 3 -valued Borel measures on the unit sphere. For 1 < p < ∞ , we prove that h e p ( B ) decomposes as the direct sum of three subspaces. The main contribution of this paper is to describe each of these subspaces along with the corresponding spaces of boundary values. In particular, two of these spaces consist of solutions of the Lamé equation for all eligible choices of the Lamé constants: one of them is the space of Riesz fields (solutions of the generalized Cauchy–Riemann equations) in h e p ( B ) ; the second is the space of fields given by the cross product of x with such Riesz fields. The results rely on the classical decomposition of L 2 vector fields on the sphere into the direct sum of three spaces of vector spherical harmonics, which we extend to L p .

PMNO: A novel physics guided multi-step neural operator predictor for partial differential equations.

Acoustic forces offer a powerful, contact-free modality for manipulating particles and fluids within microfluidic lab-on-a-chip systems. However, realizing the full potential of acoustic manipulation has been constrained by conventional cleanroom-based fabrication methods. Typically formed from high-acoustic-impedance materials like silicon or glass, these processes yield devices with limited design complexity owing to the planar channel geometries inherent in micromachining. Here, we introduce a class of polymer-based acoustofluidic platforms fabricated using micro-digital light processing (μDLP) 3D printing. In contrast to micromachining, this additive manufacturing approach enables complex, truly three-dimensional (3D) microfluidic architectures in a monolithic device form factor. We demonstrate strategies to overcome challenges associated with low-acoustic-impedance polymer resins and establish design rules based on precise control over channel and surrounding material dimensions (e.g., wall thicknesses) to achieve robust acoustofluidic functions including efficient sharp-edge-based mixing and effective particle focusing using a bulk acoustic wave resonance mode. By leveraging the design freedom provided by additive manufacturing, we fabricated an integrated, monolithic device driven by a single piezoelectric element that sequentially performs acoustic mixing and focusing within spatially distinct regions enabled by engineered variations in the 3D channel structure. This work establishes μDLP additive manufacturing as a key enabler for next generation acoustofluidic platforms by demonstrating how true 3D architectural control over channel geometry can yield integrated, multifunctional polymer acoustofluidic devices with an expanded functional design space.

The opportunity to utilize multivariate functional data types for conducting classification tasks is emerging with the growing availability of imaging data. Inspired by the extensive data provided by the Alzheimer's Disease Neuroimaging Initiative, we introduce a novel classifier tailored for multivariate functional data. Each observation in this framework may be associated with numerous functional processes, varying in dimensions, such as curves and images. Each predictor is a random element in an infinite-dimensional function space, and the number of functional predictors can potentially be much greater than the sample size. By adopting a sparse deep rectified linear unit network architecture and incorporating the LassoNet algorithm, the proposed functional Bayesian information criterion deep neural network performs feature selection and classification simultaneously, in contrast to existing functional data classifiers. This approach addresses the challenge of complex inter-correlation structures among multiple functional processes without requiring distributional assumptions. A simulation study and a real data application demonstrate its favorable performance.

Taxonomic and functional diversity patterns of nematode assemblages in a semi-enclosed coastal system (Pagasitikos gulf, Central Aegean Sea).

Renal transplantation is the preferred treatment for children with end-stage renal failure, offering superior survival, growth, and quality of life compared with long-term dialysis. Up to 40% of paediatric renal transplants are performed for congenital abnormalities of the kidneys and urinary tract (CAKUT), contrasting with adult causes such as diabetes and hypertension. Paediatric transplantation presents unique challenges due to potential donor-recipient size variation, diverse underlying pathologies, and limited availability of standardized clinical and imaging protocols. This educational review outlines the radiologist's role in the multidisciplinary management of paediatric renal transplantation, emphasizing the pre-operative and early post-operative phases. Pre-operative imaging is essential to assess vascular anatomy, bladder function, and available abdominal space; to guide surgical planning; and to minimize complications. All pre-operative imaging must be tailored to individual clinical scenarios. Post-operative imaging is performed by ultrasound with Doppler evaluation to monitor graft perfusion, vascular integrity, collecting system drainage, and perinephric complications. Awareness of normal post-operative findings, technical pitfalls, and early complications is critical. Understanding the surgical approach and anastomotic configuration enhances accurate interpretation. This review advocates for harmonization of imaging strategies across centres, to improve diagnostic accuracy and long-term outcomes in paediatric renal transplantation.

The global spectrum of plant form and function has advanced our understanding of ecological strategies, yet the role of petioles has been largely overlooked. Using 7 petiole and 6 lamina traits from 304 broadleaved woody species (2060 individuals) spanning a 4000-km transect in China, we examined the composition of petiole functional structure and its integrated trait space patterns with laminae. Petiole traits define a distinct two-dimensional functional space, structured by orthogonal axes of structural investment and hydraulic efficiency. This space is independent from, and expands beyond, the lamina economics spectrum. Temperature emerges as the dominant driver of petiole structural design and hydraulic efficiency. These findings reveal petiole as a major, independent axis of functional diversity, challenging the lamina-centric paradigm and highlighting its critical role in plant adaptation to environmental heterogeneity.

Complex spatial datasets defined on non-Euclidean domains with intricate geometry and dependence structures are fostering new frontiers in statistics. For such data, traditional spatial models developed for Euclidean settings may be inadequate, motivating approaches that explicitly account for the underlying domain geometry and locally varying dependencies. In this paper, we address the modeling of random functions over broad and complex domains by adopting low-rank spatial regression models. The underlying spatial field is represented in a reduced-dimensional subspace, obtained through a basis expansion in the corresponding functional space. These models leverage the prototypical structures of manifolds and graphs, allowing for spectral representations of the data using eigenfunctions of the graph Laplacian and the Laplace-Beltrami operator. This framework is extended to non-linear spatial regression models using a feed-forward neural networks, which can effectively handle non-Euclidean domains with complex geometries, dependencies, and manifold structures. We illustrate the proposed approach through four applications, each involving distinct types of data with increasing levels of complexity in terms of spatial domain and process characteristics. The results demonstrate the usefulness of the proposed models in accommodating intricacies of irregularly-shaped spatial domains or manifold structures.

Abstract The time-dependent Schrödinger equation (TDSE) in real space is fundamental to understanding the dynamics of many-electron quantum systems, with applications ranging from quantum chemistry to condensed matter physics and materials science. However, solving the TDSE for complex fermionic systems remains a significant challenge, particularly due to the need to capture the time-evolving many-body correlations, while the antisymmetric nature of fermionic wavefunctions complicates the function space in which these solutions must be represented. We propose a general-purpose neural network framework for solving the real-space TDSE, Fermionic Antisymmetric Spatio-Temporal Network, which treats time as an explicit input alongside spatial coordinates, enabling a unified spatiotemporal representation of complex, antisymmetric wavefunctions for fermionic systems. This approach formulates the TDSE as a global optimization problem, avoiding step-by-step propagation and supporting highly parallelizable training. The method is demonstrated on five benchmark problems: a 1D harmonic oscillator, interacting fermions in a time-dependent harmonic trap, 3D hydrogen orbital dynamics, a laser-driven hydrogen atom, and a laser-driven H$_2$ molecule, achieving excellent agreement with reference solutions across all cases. These results demonstrate the method's accuracy and flexibility within the bound-state manifold across various dimensions and interaction regimes. While the current localized Ansatz inherently restricts the description of extensive ionization and continuum states, the method demonstrates the capability to stably simulate coherent multi-electron dynamics over extended time windows. Our framework offers a highly expressive alternative to traditional basis-dependent or mean-field methods, opening new possibilities for ab initio simulations of time-dependent quantum systems, with applications in quantum dynamics, molecular control, and ultrafast spectroscopy.

Abstract We introduce a quadratic form Q on the space of functions on the gap poset G of the numerical semigroup $$\langle a,b\rangle $$ ⟨ a , b ⟩ . We prove combinatorially that when evaluated on the indicator function of an upward closed subset D , this quadratic form precisely recovers the Gorsky–Mazin $$\texttt {dinv} $$ dinv statistic of D , viewed as a Young subdiagram of G . Furthermore, we prove Theorem 1.2 that when evaluated on a pair of subdiagrams of G , the symmetric bilinear form associated with Q is equal to a novel cross- $$\texttt {dinv} $$ dinv statistic, which is non-negative. Combining these, we prove the inequality $$\begin{aligned} Q(\mathbf {\textit{n}})\ge \dfrac{1}{|G|}\,\Vert \mathbf {\textit{n}}\Vert _\infty ^2 \end{aligned}$$ Q ( n ) ≥ 1 | G | ‖ n ‖ ∞ 2 if $$\mathbf {\textit{n}}$$ n is a real-valued decreasing function on G , showing an effective positive definiteness of Q on the corresponding cone. Theorem 1.2, the main engine of the paper, was autoformalized in Lean/Mathlib by AxiomProver.

The purpose of this paper is to study the space of continuous functions, $C[a,b]$ as a $2$-normed space. In particular, we show that the space is complete with respect to some linearly independent set.

Urbanization reduces biodiversity and affects plant–insect interactions, creating a need for more functional green spaces. Urban meadows with native species are a promising option, but their use is still limited due to a variety of reasons concerning the utilization framework of suitable plant species. The present study aimed to develop seed germination protocols for 26 native Mediterranean herbaceous species originating from northeastern Greece selected to support the establishment of species-rich and self-sustaining urban meadows. To the above end, seed germination experiments were conducted ex situ under controlled environment conditions using seeds collected from the wild for each species. Seed viability was assessed using the tetrazolium (TTZ) test to determine the maximum germination potential in each case. Freshly collected seeds were stored under ambient conditions for approximately 3 months (after-ripening) prior to germination testing, which was followed by cold stratification as a pretreatment for dormancy release. The results showed high embryo viability in all species and indicated that most taxa exhibited either no dormancy or relatively shallow physiological dormancy. Germination tests revealed that 14 of the 26 species presented high germination percentages in the control treatment, which suggests that after-ripening contributed to dormancy release in a significant portion of the seed lot. However, it remains unclear whether freshly collected seeds require an initial after-ripening period before responding to cold stratification. Furthermore, cold stratification significantly enhanced germination in 12 species confirming its effectiveness as a simple and practical method for dormancy release. In addition to the seed germination results, the selected species present a wide range of functional and esthetic characteristics, including variation in plant height, flowering phenology and flower and leaf color. These traits are important for both ecological performance and visual quality in urban environments. The combination of extended flowering periods and color diversity suggests the potential for continuous floral resource availability, which can support diverse pollinator communities and, indirectly, urban fauna such as insectivorous birds. The results indicate that the studied species are suitable for biodiversity-oriented urban plantings. Their relatively shallow dormancy and ease of propagation, coupled with their functional and aesthetic traits, support their use in the development of resilient and self-sustaining urban meadows.

This paper investigates a class of third-order partial differential equations with an unbounded lower-order coefficient in the Hilbert space L2(R2). The study is motivated by the wide use of third-order equations, particularly of Korteweg–de Vries type, in mathematical physics and wave theory, as well as by the limited development of the corresponding theory in the presence of unbounded coefficients. The main focus is on the existence, uniqueness, and maximal regularity of solutions. Within a functional-analytic framework, the well-posedness of the problem is established in natural function spaces under minimal assumptions on the coefficients. In particular, a priori estimates ensuring maximal regularity are derived.

Abstract This study investigates the existence and uniqueness of solutions for a coupled system of Langevin-type fractional differential equations featuring generalized Ψ-Caputo derivatives in arbitrary Banach spaces. While prior research has predominantly focused on finite-dimensional or specific function spaces, this work extends the framework to infinite-dimensional settings, offering a more versatile analytical approach. Uniqueness is established using Banach’s fixed-point theorem under Lipschitz-type conditions, while existence is proven via Monch’s fixed-point principle combined with the measure of noncompactness—a powerful tool for infinite-dimensional problems. Demonstrative examples, including cases in the space of null sequences, validate the theoretical framework. The findings enhance the study of fractional coupled systems, introducing a flexible and comprehensive approach that integrates diverse fractional operators and strengthens foundational results.

Abstract Given an open subset U of a complex Banach space E , a weight v on U , and a complex Banach space F , let $$\mathscr {H}_v^{\infty }(U,F)$$ H v ∞ ( U , F ) denote the Banach space of all weighted holomorphic mappings $$f:U\rightarrow F$$ f : U → F , under the weighted supremum norm $$\Vert f\Vert _v:=\sup \hspace{0.55542pt}\{v(x)\Vert f(x)\Vert \,{:}\, x\in U\}$$ ‖ f ‖ v : = sup { v ( x ) ‖ f ( x ) ‖ : x ∈ U } . In this paper, we introduce and study the class $$\Pi _p^{\mathscr {H}_v^{\infty }}(U,F)$$ Π p H v ∞ ( U , F ) of p -summing weighted holomorphic mappings. We prove that it is an injective Banach ideal of weighted holomorphic mappings. Variants for weighted holomorphic mappings of Pietsch Domination Theorem, Pietsch Factorization Theorem and Maurey Extrapolation Theorem are presented. We also identify the spaces of p -summing weighted holomorphic mappings from U into $$F^*$$ F ∗ under the norm $$\pi ^{\mathscr {H}_v^{\infty }}_p$$ π p H v ∞ with the duals of F -valued $$\mathscr {H}_v^{\infty }$$ H v ∞ -molecules on U under a suitable version $$d^{\mathscr {H}_v^{\infty }}_{p^*}$$ d p ∗ H v ∞ of the Chevet–Saphar tensor norms.

Physics-informed machine learning (PIML) is emerging as a potentially transformative paradigm for modeling complex biomedical systems by integrating parameterized physical laws with data-driven methods. Here, we review three main classes of PIML frameworks: physics-informed neural networks (PINNs), neural ordinary differential equations (NODEs), and neural operators (NOs), highlighting their growing role in biomedical science and engineering. We begin with PINNs, which embed governing equations into deep learning models and have been successfully applied to biosolid and biofluid mechanics, mechanobiology, and medical imaging, among other areas. We then review NODEs, which offer continuous-time modeling, especially suited to dynamic physiological systems, pharmacokinetics, and cell signaling. Finally, we discuss deep NOs as powerful tools for learning mappings between function spaces, enabling efficient simulations across multiscale and spatially heterogeneous biological domains. Throughout, we emphasize applications where physical interpretability, data scarcity, or system complexity make conventional black-box learning insufficient. We conclude by identifying open challenges and future directions for advancing PIML in biomedical science and engineering, including issues of uncertainty quantification, generalization, and integration of PIML and large language models.

Generalized weak Galerkin methods for H(div), H(curl), and H(div, curl)-elliptic problems

Corrigendum to: “Cesàro-type operators on derivative-type Hilbert spaces of analytic functions II” [J. Funct. Anal. 290 (5) (2026) 111287

Zooplankton in tropical marine-coastal ecosystems are increasingly threatened by untreated wastewater discharges that degrade water quality and alter community structure. This study assessed these impacts in the coastal zone of Santa Marta (Colombia), comparing disturbed sectors influenced by the Manzanares River and a submarine outfall with undisturbed sectors in the Tayrona National Natural Park (Concha Bay, Neguanje Bay, and Isla Aguja) across dry and rainy seasons (March 2023-February 2024). Bimonthly sampling was conducted to characterize physicochemical parameters and their relationships with mesozooplankton taxonomic diversity and functional diversity, based on traits of body size, trophic group, feeding and spawning strategy. Untreated discharges disrupted natural physicochemical patterns, driving eutrophication marked by elevated turbidity, organic enrichment, and inorganic nutrients. It was observed that human disturbance canceled the natural seasonal pattern of diversity variation and promoted environmental filters associated with eutrophication of the system and limited the colonization of specialized and less abundant species of zooplankton communities. However, taxonomic diversity indices showed limited responses and functional metrics were more sensitive: relative functional richness declined significantly in disturbed sectors during the dry season, and functional distance revealed selective shifts in trait composition across disturbance states and seasons. Disturbances favored small-bodied, omnivorous, detritivorous, and bacterivorous taxa with ambush feeding and egg-sac reproduction, reflecting environmental filtering that homogenizes functional space and reduces resilience. In contrast, functional stability in the protected sectors highlights their role as essential reservoirs for preserving mesozooplankton integrity and providing baseline conditions for impact assessment. Our findings demonstrate that integrating taxonomic and functional approaches improves the detection of early ecological impacts and underscores the need for effective wastewater treatment to sustain biodiversity and ecosystem services in tropical coastal systems.

Monte Carlo simulation-based characteristics of indoor radon concentrations and exploratory probabilistic risk analysis in university buildings.