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.

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Recent advances in generative modeling-particularly diffusion models and flow matching-have been widely used for synthesizing discrete data such as images and videos. However, adapting these models to physical applications remains challenging, as the quantities of interest are continuous functions governed by complex physical laws. To address this, we introduce FunDiff, an efficient and robust framework for generative modeling in function spaces. FunDiff combines a latent diffusion process with a function autoencoder architecture to handle input functions with varying discretizations, generates continuous functions that can be evaluated at arbitrary locations, and seamlessly incorporate physical priors. These priors are enforced through architectural constraints or physics-informed loss functions, ensuring that generated samples satisfy fundamental physical laws. We theoretically establish minimax optimality guarantees for density estimation in function spaces, demonstrating that diffusion-based estimators achieve optimal convergence rates under suitable regularity conditions. We further demonstrate the practical effectiveness of FunDiff across diverse applications in fluid dynamics and solid mechanics. Empirical results indicate that our method can generate physically consistent samples with high fidelity to the target distribution, and exhibit robustness to noisy and low-resolution data.

Abstract Ecological communities are structured by a few common species, while most occur at low abundance. Understanding the drivers of this widespread pattern raises fundamental questions about community assembly rules and is important for applied ecology for identifying conservation targets. We used assemblages of phyllostomid bats to answer the following questions: (i) Does a higher divergence of morphological traits and functional hypervolume from the assemblage explain the prevalence of low‐abundance species? (ii) What is the relative importance of single functional traits and functional hypervolume divergence to explain such patterns? We sampled phyllostomid bats across an urban–rural landscape and estimated species abundance, measured key morphological traits, and calculated functional hypervolumes. We then applied a Bayesian causal inference framework to identify the drivers of abundance. The divergence of functional hypervolume, flight performance, and food acquisition traits had a negative impact on the abundance of the species. This pattern holds whether assessing aggregated species abundance or when considering spatiotemporal variation in assemblage structure, implying that low‐abundance species had functional hypervolumes and morphological traits more divergent from the assemblage average. Species ranked at the quantile intervals 0%–25% and 25%–50% of abundance occupied hypervolumes 45.9% and 46.8% more divergent compared to species ranked at Q 75%–100%. Similarly, the species at Q 0%–25% and 25%–50% exhibited a 59.5% greater divergence in morphological traits compared to common species. Such divergence on specific traits and intraspecific functional space from the assemblage centroid can result in a substantial reduction (13%–57%) in species abundance. Our results indicate that low‐abundance species are linked to their trait and hypervolume functional divergence. We propose that the position of the species in the functional space and the divergence of sensory‐ and vagility‐related traits are factors that determine the structure of bat communities, which denotes niche axes that have likely been narrowed at the current human‐dominated habitat. Our findings emphasize the importance of low‐abundance species, as they occupy unique ecological niches and likely contribute to specific ecosystem processes. Read the free Plain Language Summary for this article on the Journal blog.

Abstract We construct various examples of Sobolev-type functions, defined via upper gradients in metric spaces, that fail to be quasicontinuous or weakly quasicontinuous. This is done with quasi-Banach function lattices X as the function spaces defining the smoothness of the Sobolev-type functions. These results are in contrast to the case $$X=L^p$$ X = L p with $$1\le p&lt;\infty $$ 1 ≤ p &lt; ∞ , where all Sobolev-type functions in $$N^{1,p}$$ N 1 , p are known to be quasicontinuous, provided that the underlying metric space $$\mathcal {P}$$ P is locally complete. In most of our examples, $$\mathcal {P}$$ P is a compact subset of $$\textbf{R}^2$$ R 2 and $$X=L^\infty $$ X = L ∞ . Four particular examples are the damped topologist’s sine curve, the von Koch snowflake curve, the Cantor ternary set and the Sierpiński carpet. We also discuss several related properties, such as whether the Sobolev capacity is an outer capacity, and how these properties are related. A fundamental role in these considerations is played by the lack of the Vitali–Carathéodory property.

Multimodal neural operators for real-time biomechanical modelling of traumatic brain injury.

It has been known for over a century that certain large classes of functions defined on a compact nowhere dense subset X X of the complex plane, and obtained as limits of analytic functions in various metrics, can sometimes inherit the property of unique continuation characteristic of the approximating family. The first example of the transfer of the uniqueness property in this way to R ( X ) R(X) , the space of functions that can be uniformly approximated on X X by a sequence of rational functions whose poles lie outside of X X , was obtained by M. V. Keldysh around 1940, but apparently never published. Years later in 1975 A. A. Gonchar exhibited a qualitatively definitive improvement of Keldysh’s example, and our goal here is to extend that result to R p ( X , d A ) R^p(X,dA) , p ≥ 2 p\geq 2 , the evidently larger space obtained as the closure of the rational functions in L p ( X , d A ) L^p(X,dA) , where d A dA denotes 2 2 -dimensional Lebesgue, or area, measure.

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

Given a positive Borel measure μ \mu on [ 0 , 1 ) [0,1) and a parameter β &gt; 0 \beta &gt;0 , we consider the Cesàro-type operator C μ , β \mathcal {C}_{\mu ,\beta } acting on the analytic function f ( z ) = ∑ n = 0 ∞ a n z n f(z)=\sum _{n=0}^{\infty } a_{n} z^{n} on the unit disc of the complex plane D \mathbb {D} , defined by C μ , β ( f ) ( z ) = ∑ n = 0 ∞ μ n ( ∑ k = 0 n Γ ( n − k + β ) ( n − k ) ! Γ ( β ) a k ) z n = ∫ 0 1 f ( t z ) ( 1 − t z ) β d μ ( t ) , \begin{equation*} \mathcal {C}_{\mu ,\beta }(f)(z)= \sum _{n=0}^{\infty } \mu _{n} \left ( \sum _{k=0}^{n} \frac {\Gamma (n-k+\beta )}{(n-k)! \Gamma (\beta )} a_{k} \right ) z^{n} = \int _{0}^{1} \frac {f(tz)}{(1-tz)^{\beta }} d\mu (t), \end{equation*} where μ n = ∫ 0 1 t n d μ ( t ) \mu _{n}=\int _{0}^{1} t^{n} d\mu (t) . This operator generalizes the classical Cesàro operator (corresponding to the case where μ \mu is the Lebesgue measure and β = 1 \beta =1 ) and includes other relevant cases previously studied in the literature. In this paper we study the boundedness of C μ , β \mathcal {C}_{\mu ,\beta } on mixed norm spaces H ( p , q , γ ) H(p,q,\gamma ) for 0 &gt; p , q ≤ ∞ 0&gt;p,q\leq \infty and γ &gt; 0 \gamma &gt;0 . Our results extend and unify several known characterizations for the boundedness of Cesàro-type operators acting on spaces of analytic functions.

Abstract We study the numerical computation of nontrivial critical points of variational functionals associated with nonlinear Dirichlet problems involving the p -Laplacian. Previous numerical mountain pass approaches typically relied on finite element discretizations of the underlying function space. In contrast, we employ a discretization based on a geometric B-spline representation of the solution. The function space is approximated by smooth spline curves parameterized by control points, yielding a finite-dimensional geometric representation of the variational problem. Within this discrete space we apply a mountain-pass type up–down method. This allows the search for saddle-type critical points to be carried out directly in the space of spline control points. The descent direction is obtained through an auxiliary Poisson equation, providing a Sobolev gradient that stabilizes the iteration. Convergence of the numerical procedure is monitored via the Euler–Lagrange residual, ensuring that the computed spline approximation satisfies the variational problem up to a prescribed tolerance. Numerical experiments for the model case $$p=2$$ p = 2 with nonlinearity $$f(u)=u^3$$ f ( u ) = u 3 on $$\Omega =(0,1)$$ Ω = ( 0 , 1 ) show that the method computes nontrivial solutions, including sign-changing profiles depending on the initialization. The results demonstrate the successful application of B-splines in the numerical solution of nonlinear variational problems.

In this paper, we investigate a (k,Υ) fractional quadratic integral equation in the Banach space of real-valued continuous functions on [0,1]. By using a measure of noncompactness associated with monotonicity and Darbo’s fixed point theorem, we provide sufficient conditions for the existence of at least one monotonic solution and analyze its stability. Finally, an illustrative example is presented to demonstrate the theoretical results, including several particular cases.

We introduce a method for the fast numerical approximation of linear, second-order parabolic partial differential equations (PDEs for short) with time-independent coefficients based on model order reduction techniques and the Laplace transform. We start by applying this transform to the evolution problem, thus yielding a time-independent boundary value problem solely depending on the complex Laplace variable. In an offline stage, we judiciously sample the Laplace variable and numerically solve the corresponding collection of high-fidelity or full-order problems. Next, we apply a proper orthogonal decomposition (POD) to this collection of solutions in order to obtain a reduced basis in the Laplace domain. We project the linear parabolic problem onto this basis and then, using any suitable time-stepping method, we solve the evolution problem. A key insight to justify the implementation and analysis of the proposed method consists of using Hardy spaces of analytic functions and establishing, through the Paley-Wiener theorem, an isometry between the solution of the time-dependent problem and its Laplace transform. As a result, one may conclude that computing a POD with samples taken in the Laplace domain produces an exponentially accurate reduced basis for the time-dependent problem. Numerical experiments illustrate the performance of the method in terms of accuracy and, in particular, speed-up when compared to the solution obtained by solving the full-order model.

ABSTRACT Conflict and defense strategies significantly shape the evolution of settlement space. This study examines the Tunpu settlements in central Guizhou, China, as a case example to investigate the spatial adaptation mechanisms of military defense heritage within conflict-affected environments. By integrating historical documents, GIS analysis, and field investigations, the research identifies three adaptive characteristics that emerged during the transition of Tunpu from a state-led military system to a community-based self-defense structure: first, the strategic use of karst terrain to create a “natural-artificial” composite defense system, reflecting the ecological wisdom of “low-tech, high-intelligence” design; second, the coexistence of multi-ethnic architectural techniques fostering the secularization of functional spaces; and third, the dynamic transmission of military heritage through cultural hybridity. A comparative analysis with international defensive settlements reveals that the resilience of Tunpu settlements resides in a community-led mechanism of functional continuity and spatial reproduction. Building on these findings, this paper proposes an adaptive cycle conservation strategy to support the dynamic inheritance and sustainable development of cultural heritage.

In this paper, we rigorously analyse a linear elastic isotropic rod model of the Naghdi type. The model is formulated in the functional space H 1 , without additional constraints in the function space and well defined for W 1 , ∞ parametrisation of the middle line. We prove the model’s mathematical well-posedness. The model is justified through a detailed asymptotic analysis, showing that its solution converges to the same limit as the solution of the three-dimensional (3D) equations of linearised elasticity as the thickness parameter ( h ) tends to zero. The proposed model is compared, both analytically and numerically, against the flexural rod model and the full 3D elasticity equations.

Major neuropsychiatric disorders such as major depressive disorder (MDD) and schizophrenia (SCZ), as well as the neurodevelopmental disorder autism spectrum disorder (ASD), are traditionally treated as distinct clinical entities. However, genome-wide association studies indicate shared genetic risks, motivating a transdiagnostic view. Resting-state functional connectivity (rsFC) is a promising biomarker for these disorders, but its high dimensionality complicates inference of inter-disorder relationships in the native feature space. Here, we develop an rsFC-based embedding-relation workflow that quantifies disorder relationships in a connectivity-informed, low-dimensional embedding space. Central to the workflow is a mutual information-based embedding framework that evaluates candidate embedding approaches and selects an optimal strategy. Using synthetic connectivity data, the framework indicates that rsFC embeddings are best represented in a spherical space under a moderate level of supervision. Building on this insight, we applied the workflow to curated, multi-disorder rsFC datasets to derive shared embedding spaces encompassing the connectivity features of ASD, MDD, and SCZ. In these spaces, we consistently observed a robust three-way relationship: a pronounced neurobiological dissimilarity between ASD and MDD, contrasted with greater similarity between SCZ and both disorders. These findings support a dimensional, transdiagnostic perspective on neuropsychiatric disorders and offer new insights into their shared and distinct neural underpinnings.

As compared with large-scale infrastructures with much attention in the design, construction, and operation, there are a huge number of small-scale infrastructures with a relatively lower safety factor and under deficient conditions. These structures rarely own a complete structural health monitoring (SHM) system due to budget limitations. Given the fact that the neighbored infrastructures are subjected to correlated load effects, their responses are spatiotemporally correlated. In this study, the concept of neighbored SHM is thereby proposed for the neighbored small-scale infrastructures under a sparse, lightweight, and data-driven sensing framework. The cross-correlation coefficients (CCCs) of the responses from multiple neighbored structures are theoretically derived, and the associated probability space is subsequently modeled. An anomaly can be detected by estimating the deviations of CCCs from their baselines. A decision matrix is finally obtained to identify the abnormal elementary structure from a set of neighbored structures, where installing only a single sensor is required on each elementary structure. The proposed approach has a sound theoretical basis, light computational cost, and low implementing burdens; thus, it applies to small-scale infrastructures with large populations. Numerical and laboratory experiments are conducted using the mass-spring systems and simply-supported beams. Results have demonstrated that the developed neighbored SHM strategy is efficient and sensitive to structural anomaly.

The local diameter two property and the diameter two property in spaces of Lipschitz functions

Biochemical reactions are inherently stochastic, with their kinetics commonly described by chemical master equations (CMEs). However, the discrete nature of molecular states renders likelihood-based parameter inference from CMEs computationally intensive. Here, we introduce an inference method that leverages analytical solutions in the probability generating function (PGF) space and systematically evaluate its efficiency, accuracy, and robustness. Across both steady-state and time-resolved count data, our numerical experiments demonstrate that the PGF-based method consistently outperforms existing approaches in terms of both computational efficiency and inference accuracy, even under data contamination. These favorable properties further enable the extension of the PGF-based framework to model selection-a task typically considered computationally prohibitive. Using time-resolved data, we show that the method can correctly identify complex gene expression models with more than three gene states, a task that cannot be reliably achieved using steady-state data alone.