We propose a containment query that is robust to the watertightness of regions bound by trimmed NURBS surfaces, as this property is difficult to guarantee for in-the-wild CAD models. Containment is determined through the generalized winding number (GWN), a mathematical construction that is indifferent to the arrangement of surfaces in the shape. Applying contemporary techniques for the 3D GWN to trimmed NURBS surfaces requires some form of geometric discretization, introducing computational inefficiency to the algorithm and even risking containment misclassifications near the surface. In contrast, our proposed method leverages properties of the 3D solid angle to solve the relevant surface integral using a boundary formulation with rapidly converging adaptive quadrature. Batches of queries are further accelerated by memoizing (i.e. caching and reusing) quadrature node positions and tangents as they are evaluated. We demonstrate that our GWN method is robust to complex trimming geometry in a CAD model, and is accurate up to arbitrary precision at arbitrary distances from the surface. The derived containment query is therefore robust to model non-watertightness while respecting all curved features of the input shape.

Carbon fiber-reinforced plastics (CFRPs) have become increasingly significant in recent decades due to their remarkable mechanical properties and lightweight nature. This study aims to advance the understanding and simulation of CFRP behavior through the development of a hyperelastic-plastic-damage homogenization method combined with mean-field theory. The material responses of both the fiber and matrix are modeled using strain energy functions that account for damage evolution, while a complete linearization of the homogenization process is derived to ensure the consistent implementation of the Newton–Raphson iteration scheme in large deformation simulations. The innovative aspect of this work lies in the constitutive linearization for the hyperelastic-plastic-damage formulation within a mean-field homogenization framework, providing an efficient Newton algorithm for modeling the nonlinear behavior of CFRP. A failure criterion for the hyperelastic model of fibers is introduced, along with a damage saturation variable in rate form for the matrix, effectively capturing damage evolution. Through discrete formulations for the homogenization, the proposed model’s capability is demonstrated via three numerical examples and validated against experimental investigations, proving its effectiveness and reliability in simulating CFRP damage.

Protein loop modeling remains a fundamental challenge in computational biology due to the inherent flexibility of loops and their critical role in biological functions. In this work, we employ a discrete distance geometry formulation, efficiently solved using the Branch-and-Prune algorithm, with a key innovation being the incorporation of hydrogen atoms into the model. Hydrogen atoms bonded to N and C α in the protein backbone introduce additional geometric constraints, and their inclusion is particularly justified in the context of nuclear magnetic resonance (NMR) experiments, where short-range hydrogen-hydrogen distances can be detected and provide valuable structural information. By integrating these experimentally accessible constraints into the modeling process, we refine the representation of protein conformations. Computational experiments demonstrate that incorporating hydrogen atoms reduces the conformational space, leading to a more constrained and biologically realistic model. Comparisons with hydrogen-free formulations confirm that our approach improves agreement with known protein structures, further highlighting the relevance of distance geometry methods in structural refinement.

This paper develops a unified fractional version of the Hermite–Hadamard inequality and Bullen-type inequalities for convex functions defined on discrete time scales. By employing generalized fractional difference operators, the obtained result encompasses and extends previously known discrete formulations, including both the classical case and higher-order variants. Furthermore, we investigate the approximation accuracy of the introduced fractional mean operator. Specifically, we establish explicit error bounds for Lipschitz functions and for functions with convex differences, providing a more comprehensive analysis of the discrete fractional setting.

This article offers a comprehensive examination of the role of artificial intelligence (AI) technologies in the transformation of the public administration system under conditions of digitalisation of the public sector. It analyses the principal institutional challenges faced by public authorities in the implementation of algorithmic solutions, including the need to modernise organisational structures, develop the professional competencies of civil servants, and ensure transparency, accountability, and ethical justification of automated administrative procedures. Particular attention is paid to the evolution of managerial practices, manifested in a gradual shift from traditional administrative procedures towards data-centric, predictive, and algorithmically supported models of decision-making. It is argued that the integration of artificial intelligence leads to the emergence of new forms of administrative discretion, alters the balance between human and automated influence within governance processes, and exerts a significant impact on the structure of public policy, performance evaluation mechanisms, and the quality of public service delivery. From a regulatory and legal perspective, the article examines contemporary European approaches to the governance of artificial intelligence, including requirements related to safety, transparency, non-discrimination, and the protection of human rights. The significance of European Union legal acts, ethical principles developed by the Council of Europe, and recommendations of international organisations is outlined, as these instruments establish framework standards for the responsible use of algorithmic systems in the public sector. The article substantiates the necessity of developing a national regulatory model capable of reconciling technological innovation, administrative efficiency, and guarantees of democratic legitimacy. It is concluded that the effectiveness of digital transformation in public administration depends not solely on the technological capacity of artificial intelligence, but primarily on the institutional capability of the state, the quality of legal regulation, and the ability to ensure an appropriate level of public trust in algorithmically supported administrative decisions, including through the implementation of risk assessment procedures, oversight mechanisms, and accountability frameworks governing the use of artificial intelligence.

In this paper, we formulate, analyse and implement the discrete formulation of the Brinkman problem with mixed boundary conditions, including slip boundary condition, using the Nitsche’s technique for virtual element methods. We propose a discretization by means of the virtual elements presented in [9]. We derive a robust stability analysis of the Nitsche stabilized discrete scheme for this model problem. We establish optimal a priori error estimates of the discrete scheme with constants independent of the viscosity. Moreover, a set of numerical tests demonstrates the robustness with respect to the physical parameters and verifies the derived convergence results.

We develop and analyze an adaptive spacetime finite element method for nonlinear parabolic equations of p–Laplace type. The model problem is governed by a strongly nonlinear diffusion operator that may be degenerate or singular depending on the exponent p, which typically leads to limited regularity of weak solutions. To address these challenges, we formulate the problem in a unified spacetime variational framework and discretize it using conforming finite element spaces defined on adaptive spacetime meshes. We prove the well-posedness of both the continuous problem and the spacetime discrete formulation, and establish a discrete energy stability estimate that is uniform with respect to the mesh size. Based on residuals in the spacetime domain, we construct a posteriori error estimators and prove their reliability and local efficiency. These results form the foundation for an adaptive spacetime refinement strategy, for which we prove global convergence and quasi-optimal convergence rates without assuming additional regularity of the exact solution. Numerical experiments confirm the theoretical findings and demonstrate that the adaptive spacetime finite element method significantly outperforms uniform refinement and classical time-stepping finite element approaches, particularly for problems exhibiting localized spatial and temporal singularities.

Abstract A discrete mechanical model for large-amplitude free vibrations of two-stepped functionally graded beams is developed in this study. The beam’s material properties vary through the thickness according to a power-law distribution between the metallic and ceramic phases. The continuous beam is replaced by an N-degree-of-freedom system of lumped masses, longitudinal, and torsional springs. Using Hamilton’s principle, the governing nonlinear algebraic equations are derived and solved through the single-mode approach (SMA) to obtain the nonlinear frequency-amplitude relationships. In addition, an Artificial Neural Network (ANN)-based surrogate model is proposed to provide fast and accurate predictions of the nonlinear-to-linear frequency ratio as a function of key parameters such as step ratio, step position, boundary conditions, and the power-law index. Trained on data generated by the discrete formulation, the surrogate attains excellent generalization with a drastic reduction in computation time. The combined discrete-ANN framework offers both physical interpretability and computational efficiency, making it suitable for rapid design and optimization of complex FGM beam structures.

In translationally invariant semiconductors that host exciton bound states, one can define an infinite number of possible exciton Berry connections. These correspond to the different ways in which a many-body exciton state, at fixed total momentum, can be decomposed into free electron and hole Bloch states that are entangled by an exciton envelope wave function. Inspired by the modern theory of polarization, we define an exciton projected position operator whose eigenvalues single out two unique choices of exciton Berry phase and associated Berry connection—one for electrons, and one for holes. We clarify the physical meaning of these exciton Berry phases and provide a discrete Wilson loop formulation that allows for their numerical calculation without a smooth gauge. As a corollary, we obtain a gauge-invariant expression for the at a given total momentum, i.e., the mean separation of the electron and hole within the exciton wave function. In the presence of crystalline inversion symmetry, the electron and hole exciton Berry phases are quantized to the same value and we derive how this value can be expressed in terms of inversion eigenvalues of the many-body exciton state. We then consider C 2 T symmetry, for which no symmetry eigenvalues are available as it is antiunitary, and confirm that the exciton Berry phase remains quantized and still diagnoses topologically distinct exciton bands. The notion of shift excitons, whose exciton Wannier states are displaced from those of the noninteracting bands by a quantized amount, can therefore be generalized beyond symmetry indicators.

Abstract The dynamics of a hanging chain pendulum, long treated as a textbook problem in classical mechanics, are revisited from a fresh and rigorous analytical perspective. By systematically deriving and comparing the continuum and discrete formulations, subtle but significant differences in the vibrational spectrum, particularly in the high-frequency regime are uncovered. Using asymptotic expansions, boundary layer theory, and matched scaling arguments, a comprehensive description of the eigenmodes and their scaling behavior is developed. In the discrete model, we reveal a striking two-regime structure: low-frequency modes governed by Bessel-type equations, and high-frequency modes localized near the free end, described by Airy-type asymptotics. The transition between these regimes emerges naturally from a balance of competing terms in the governing equations, yielding a characteristic crossover scaling. This analysis clarifies the limitations of discrete and continuum approximations and exposes the deeper mathematical structure underlying the system. Ultimately, the followed approach provides a dual perspective and case study, demonstrating how rigorous asymptotics bridge discrete and continuum models and yield fresh insight into seemingly well-understood mechanics of the chain pendulum.

Graph clustering is a longstanding topic in machine learning. In recent years, deep learning methods have achieved encouraging results, but they still require predefined cluster numbers $K$, and typically struggle with imbalanced graphs, especially in identifying minority clusters. The limitations motivate us to study a challenging yet practical problem: deep graph clustering without $K$ considering the imbalance in reality. We approach this problem from a fresh perspective of information theory (i.e., structural information). In the literature, structural information has rarely been touched in deep clustering, and the classic definition falls short in its discrete formulation, neglecting node attributes and exhibiting prohibitive complexity. In this paper, we first establish a differentiable structural information, generalizing the discrete formalism to continuous realm, so that we design a hyperbolic deep model (LSEnet) to learn the neural partitioning tree in the Lorentz model of hyperbolic space. Theoretically, we demonstrate its capability in clustering without requiring $K$ and identifying minority clusters in imbalanced graphs. Second, we refine hyperbolic representations of the partitioning tree, enhancing graph semantics, for better clustering. Contrastive learning for tree structures is non-trivial and costs quadratic complexity. Instead, we further advance our theory by discovering an interesting fact that structural entropy indeed bounds the tree contrastive loss. Finally, with an efficient reformulation, we approach graph clustering through a novel augmented structural information learning (ASIL), which offers a simple yet effective objective of augmented structural entropy to seamlessly integrates hyperbolic partitioning tree construction and contrastive learning. With a provable improvement in graph conductance, ASIL achieves effective debiased graph clustering in linear complexity with respect to the graph size. Extensive experiments show the ASIL outperforms 20 strong baselines by an average of $+12.42\%$ in NMI on Citeseer dataset.

Objective: This study investigates the hydrothermal characteristics of water within the annular region of a thermal energy storage (TES) system featuring a wavy shell wall and a cylindrical tube equipped with three fins. The primary objectives are to analyze the impact of cylinder position and rotation on flow dynamics and thermal behavior, assess entropy generation under an inclined magnetic field and porous medium effects, and evaluate the role of copper nanoparticles in enhancing system performance. Methodology: The problem is modeled using dimensional governing equations and transformed into a coupled dimensionless partial differential framework. Numerical simulations are conducted via the finite element method in COMSOL, employing hybrid meshing and weak formulation for domain discretization. Grid independence tests and validation benchmarks ensure computational accuracy. Parametric variations in cylinder orientation, magnetic field inclination, and nanoparticle inclusion are systematically explored. Key Findings: Cylinder rotation and positional adjustments significantly influence velocity profiles and temperature gradients, and the inclined magnetic field amplifies flow resistance. Copper nanoparticles improve thermal conductivity, elevating the average Nusselt number by up to 18%. Entropy analysis reveals that thermal irreversibility dominates viscous effects, with Bejan number trends highlighting optimized configurations for minimizing energy losses. Applications: The findings demonstrate that TES systems with wavy geometries, finned tubes, and nanoparticle-enhanced fluids can achieve superior thermal efficiency and reduced entropy generation. This work provides actionable insights for designing advanced thermal storage solutions, particularly in renewable energy applications where intermittent sources like solar energy require robust heat management strategies.

This paper presents an advanced six-step linear multistep method of order five, based on second derivative non-hybrid block backward differentiation formula (BDF), developed for numerical solution of stiff systems of ordinary differential equations. The development of the multistep collocation approach is carried out using the matrix inversion techniques. The block structure facilitates simultaneous computation of multiple solution points, improving computational efficiency. Moreover, the power series is adopted as basis function for driving the discrete and continuous formulations. The analysis of the method such as consistency, zero-stability, order and error constants is presented, confirming its suitability for stiff systems. Numerical experiments on standard are tested on stiff and non-stiff ordinary differential equations showed the new method outperforms existing method in terms of accuracy.

Curve reconstruction is the process of estimating a smooth function or curve that fits a given setof data points, either exactly (interpolation) or approximately (fitting). Classical approaches,including global polynomial interpolation, splines, Hermite interpolation, and radial basisfunction fitting, face challenges when data are sparse, irregularly distributed, or noisy. In thispaper, we propose a curve reconstruction method based on the discrete form of the biharmonicequation. The method formulates reconstruction as a constrained quadratic optimizationproblem, incorporating both equality and inequality constraints and producing globallyC1smooth curves. The approach is physically interpretable, penalizing excessive bending, as in thecase of a thin elastic beam, and can be extended to higher-dimensional surface reconstruction.Performanceisevaluatedthroughnumericalexperimentsonknownfunctionsandsyntheticdatawith various distributions and constraints, including small perturbation tests to assess stabilityand robustness. The results demonstrate that the proposed method reproduces the data, enforcesthe prescribed bounds, and remains stable under irregular sampling and noise.

Based on a decoupling assumption made for approximating the wheelset responses, this paper presents a physically meaningful wheel-rail force model along with an efficient decoupled method for random vibration analysis of the train-bridge interaction problems. Specifically, the wheel-rail forces are treated as a multi-variate stationary random process statistically characterized by its spectral density matrix, obtained through stochastic analysis of the train equation incorporating only the track irregularity while excluding the train gravity. Next, the obtained spectral density matrix is decomposed through the spectral representation technique, yielding a discrete formulation of the wheel-rail forces with respects to a pair of orthogonal stochastic vectors (OSVs). Then, the Duhamel integral is employed to derive explicit relations between the bridge response and the OSVs, with the associated coefficient matrices efficiently determined using a recursive scheme derived based on the Newmark integration algorithm. The resulting response-OSVs expression allows for statistical calculation of the bridge responses in a decoupled manner, avoiding iterative computations or repeated system matrix updates typically required in coupled stochastic analyses of train-bridge systems. Lastly, the proposed method is numerically demonstrated through stochastic analysis of a three-span continuous bridge subjected to train loads, with comparisons made to the Monte Carlo simulation, pseudo-excitation method and a fully coupled approach. The numerical results suggest that, compared with traditional coupled approach, the proposed decoupled strategy improves significantly the computational efficiency, particularly for cases involving a large number of vehicles.

We demonstrate that, by utilizing the boundary conformal field theory (BCFT) operator algebra of the Liouville CFT, one can express its path-integral on any Riemann surface as a three dimensional path-integral with appropriate boundary conditions, generalising the recipe for rational CFTs. This serves as a constructive method for deriving the quantum holographic dual of the CFT, which reduces to Einstein gravity in the large central charge limit. As a byproduct, the framework provides an explicit discrete state-sum of a 3D non-chiral topological theory constructed from quantum 6j 6 j symbols of \mathcal{U}_q(sl(2,\mathbb{R})) 𝒰 q ( s l ( 2 , ℝ ) ) with non-trivial boundary conditions, representing a long-sought non-perturbative discrete formulation of 3D pure gravity with negative cosmological constant, at least within a class of three manifolds. This constitutes the first example of an exact holographic tensor network that reproduces a known irrational CFT with a precise quantum gravitational interpretation.

This article examines the hierarchical organization of English syntax within generative grammar, with particular emphasis on noun phrase groups and phrasal structure. It examines how words are combined into phrases and sentences, as well as analyzes the relationship between deep and surface structures and the syntaxic roles ¬ the components. On the material of both academic discourse and ¬ language, structural diagnostic methods are used in the work - such as for ¬ installation, movement and attachment - to identify hidden syntactic laws ¬ numbers. Particular attention is paid to the internal structure of the English noun phrase, including the interaction of vertex, modifiers and specifiers, the difference between mandatory and optional elements and their presentation using cons ¬ tituent structure rules. The analysis emphasizes the functions of determinatives, adjectival and prepositional groups, as well as the theoretical motivation of the intermediate kate ¬ the N-bar mountain (N ′). Integrating the fundamental ideas of N. Chomsky (3, 4, 5, 6) with more recent developments in ¬ field of generative syntax, the article demonstrates how the hierarchical structure explains distributive restrictions, interpretive differences and variability of complexity in different registers, clarifying the relationship between syntak ¬ sic form and meaning. Name groups (NP) constitute the central unit of English syntax, acting as the main carriers of the reference value. The article examines the structural components of nominal groups, their syntactic functions and manifestations of discontinuous (discrete) forms. Particular attention is paid to the interaction of vertices, determinatives, modifiers and complements, as well as the functional ¬ of name groups in written and oral registers.

The paper contains the spectral study of Laplacians that are broken on multidimensional fractal domains with a particular focus on the convergence, stability, and localization phenomena. Self-similar structures in fractal geometry were built in two and three dimensions by iterating a system of functions, contraction mappings, and by means of self-similarity parameters that are carefully determined. In the study, the broken Laplacian operator, a Laplacian operator obtained by piecewise Laplacians and domain decomposition methods, has been analyzed analytically and numerically using graph approximations and discrete energy forms. The spectral problem was posed in both the classical and variational form, and the eigenvalues and eigenfunctions were calculated at the refinement levels, showing smooth convergence towards relative errors of less than 0.3 percent to the high-resolution approximations. The fractal domain was broken through controlled discontinuities, and the eigenvalues shifted measurably with increasingly large fracture parameters of 12% and larger eigenvalues changing up to 31.4%. Eigenfunction localization was noted as well, where 27 percent of the total L^2 -mass was found to be concentrated in less than 6 percent of the domain, which confirms the sensitivity of high-frequency modes to geometric irregularities. The reduction of spectral dimensions by 8.55% and lifting of degeneracies caused by symmetry under anisotropic fragmentation was found through comparative analysis with classical Laplacians. These findings support the methodological scheme and indicate that broken Laplacians are an effective and mathematically consistent way to describe spectral properties of complex fractal spaces, and that they can be applied in the modeling of diffusion, wave propagation, and energy localization in heterogeneous media.

Abstract The optimal $L^{2}$-error estimates of a locking-free numerical method are established for a quasi-static linear poroelasticity. Existing research shows that the original model can be reformulated as a generalized Stokes equation coupled with a diffusion problem, which inherently mitigates two locking phenomena associated with the continuous Galerkin method. However, previous studies only achieved optimal error estimates in the $H^{1}$-norm, lacking results for displacement in the $L^{2}$-norm. We find that the main barrier to achieve these estimates is arising from the influence of Lamé constant $\lambda $ on numerical schemes. By taking the value of $\lambda $ into account, we design a fully discrete mixed finite element method to solve the reformulated problem. We prove that the discrete formulation satisfies the inf-sup stability condition across various finite element pairs. Our results show that when $\lambda <\infty,$ the generalized Stokes problem is stable with equal-order Lagrange element pairs, making the Taylor–Hood pairs unnecessary and enabling optimal $L^{2}$-error estimates for displacement. For $\lambda \rightarrow \infty,$ the problem reduces to a standard Stokes problem. Using inf-sup bounds and the Aubin–Nitsche duality method, we provide optimal $L^{2}$-error analysis. Numerical examples are included to confirm our theoretical findings.

Abstract In this work we present an a priori error analysis for solving the unsteady advection equation on cut cell meshes along a straight ramp in two dimensions. The space discretization uses a lowest order upwind-type discontinuous Galerkin scheme involving a Domain of Dependence (DoD) stabilization to correct the update in the neighborhood of small cut cells. Thereby, it is possible to employ explicit time stepping schemes with a time step length that is independent of the size of the very small cut cells. Our error analysis is based on a general framework for error estimates for first-order linear partial differential equations that relies on consistency, boundedness, and discrete dissipation of the discrete bilinear form. We prove these properties for the space discretization involving DoD stabilization. This allows us to prove, for the fully discrete scheme, a quasi-optimal error estimate of order one half in a norm that combines the $$L^\infty $$ L ∞ -in-time $$L^2$$ L 2 -in-space norm and a seminorm that contains velocity weighted jumps. We also provide corresponding numerical results.