Search Results (771)

The primary objective of this work is to develop a comprehensive theory of weighted fractional Sobolev spaces within the framework of timescales. To this end, we first introduce a novel class of weighted fractional operators and rigorously define associated weighted integrable spaces on timescales, generalising classical notions to this non-uniform temporal domain. Building upon these foundations, we systematically investigate the fundamental functional-analytic properties of the resulting Sobolev spaces. Specifically, we establish their completeness under appropriate norms, prove reflexivity under appropriate duality pairings, and demonstrate separability under mild conditions on the weight functions. As a pivotal application of our theoretical framework, we derive two robust existence theorems for solutions to the proposed model. These results not only extend classical partial differential equation theory to timescales but also provide a versatile tool for analysing dynamic systems with heterogeneous temporal domains. Full article

Studying the Bernstein approximations for fuzzy functions is a new attempt. With the help of considering the support functions of fuzzy sets in which the weak* topology for the normed dual space is involved, we are able to approximate continuous fuzzy functions by considering the Bernstein polynomials for fuzzy functions. We first study the Bernstein approximations for the support functions of fuzzy sets. Using the concept of isometry between the metric spaces of fuzzy sets and the normed spaces of support functions of fuzzy sets, the Bernstein approximations for the support functions of fuzzy sets can naturally lead to the Bernstein approximations for continuous fuzzy functions. Full article

Semantic modeling of legal reasoning is an important research direction in the field of artificial intelligence and law (AI and law), aiming to enhance judicial transparency, fairness, and the consistency of legal applications through structured semantic representations. This paper proposes a semantic judicial reasoning framework based on the “Data–Information–Knowledge–Wisdom–Purpose” (DIKWP) model, which transforms the conceptual expressions of traditional legal judgment into DIKWP graphs enriched with semantics. The framework integrates the objective content of legal norms with stakeholders’ subjective cognition through a DIKWP×DIKWP bidirectional mapping mechanism, achieving “semantic justice”. Specifically, we define a DIKWP-based legal knowledge representation method and design a mapping algorithm from traditional legal concepts to the DIKWP semantic structure. To validate the effectiveness of the framework, we use a real administrative law case as an example and construct DIKWP (normative content) and DIKWP (subjective cognition) graphs to model legal rules, evidence, and various perspectives. The results indicate that the intention-driven semantic transformation mechanism can harmonize legal reasoning with stakeholders’ cognitive backgrounds, thereby enhancing the interpretability and fairness of judicial interpretation. Case analysis further demonstrates that reasoning within the DIKWP semantic space can reveal underlying assumptions, bridge cognitive gaps, and promote judicial fairness by aligning legal intentions. This study provides new theoretical and methodological support for the explainable reasoning of intelligent judicial systems. Full article

The COVID-19 pandemic’s rapid shift to working from home has fundamentally challenged residential acoustic design, which traditionally prioritises rest and relaxation rather than sustained concentration. However, a clear gap exists in understanding how acoustic needs and the subjective evaluation of soundscape appropriateness (SA) differ between these conflicting activities within the same domestic space. Addressing this gap, this study reveals critical differences in how people experience and evaluate home soundscapes during work versus relaxation activities in the same residential spaces. Through an online survey of 247 Chinese participants during lockdown, we assessed soundscape perception attributes, the perceived saliencies of various sound types, and soundscape appropriateness (SA) ratings while working and relaxing at home. Our findings demonstrate that working at home creates a more demanding acoustic context: participants perceived indoor soundscapes as significantly less comfortable and less full of content when working compared to relaxing (p < 0.001), with natural sounds becoming less noticeable (−13.3%) and distracting household sounds more prominent (+7.5%). Structural equation modelling revealed distinct influence mechanisms: while comfort significantly mediates SA enhancement in both activities, the effect is stronger during relaxation (R² = 0.18). Critically, outdoor man-made noise, building-service noise, and neighbour sounds all negatively impact SA during work, with neighbour sounds showing the largest detrimental effect (total effect size = −0.17), whereas only neighbour sounds and outdoor man-made noise significantly disrupt relaxation activities. Additionally, natural sounds act as a positive factor during relaxation. These results expose a fundamental mismatch: existing residential acoustic environments, designed primarily for rest, fail to support the cognitive demands of work activities. This study provides evidence-based insights for acoustic design interventions, emphasising the need for activity-specific soundscape considerations in residential spaces. As hybrid work arrangements become the norm post-pandemic, our findings highlight the urgency of reimagining residential acoustic design to accommodate both focused work and restorative relaxation within the same home. Full article

In the present work, we give a new proof of the well-known Selberg’s inequality in complex Hilbert spaces from an operator-theoretic perspective, establishing its fundamental equivalence with the Cauchy–Bunyakovsky–Schwarz inequality. We also derive several lower and upper bounds for the Selberg operator, including its norm estimates, refining classical results such as de Bruijn’s and Bohr’s inequalities. Additionally, we revisit a recent claim in the literature, providing a clarification of the conditions under which Selberg’s inequality extends to abstract bilinear forms. Full article

Suppose X and Y are Banach spaces, K is a compact Hausdorff space, and $C(K, X)$ is the Banach space of all continuous X-valued functions (with the supremum norm). We will study some strongly bounded operators $T:C(K, X)\to Y$ with representing measures $m:\Sigma \to L(X,Y)$, where $L(X,Y)$ is the Banach space of all operators $T:X\to Y$ and $\Sigma$ is the $\sigma$-algebra of Borel subsets of K. The classes of operators that we will discuss are the Grothendieck, p-limited, p-compact, limited, operators with completely continuous, unconditionally converging, and p-converging adjoints, compact, and absolutely summing. We give a characterization of the limited operators (resp. operators with completely continuous, unconditionally converging, p-convergent adjoints) in terms of their representing measures. Full article

Using the Dixmier angle between two closed subspaces of a complex Hilbert space $\mathcal{H}$, we establish the necessary and sufficient conditions for the operator norm of the sum of two orthogonal projections, $P_{{\mathcal{W}}_1}$ and $P_{{\mathcal{W}}_2}$, onto closed subspaces ${\mathcal{W}}_1$ and ${\mathcal{W}}_2$, to attain its maximum, namely $\parallel P_{{\mathcal{W}}_1}+P_{{\mathcal{W}}_2}\parallel =2.$ These conditions are expressed in terms of the geometric relationship and symmetry between the ranges of the projections. We apply these results to orthogonal projections associated with a closed-range operator via its Moore–Penrose inverse. Additionally, for any bounded operator T with closed range in $\mathcal{H}$, we derive sufficient conditions ensuring $\parallel T{T}^{†}+T^{†}T\parallel =2,$ where $T^{†}$ denotes the Moore–Penrose inverse of T. This work highlights how symmetry between operator ranges and their algebraic structure governs norm extremality and extends a recent finite-dimensional result to the general Hilbert space setting. Full article

This study presents an image inpainting model based on an energy functional that incorporates the $L^p$ norm of the fractional Laplacian operator as a regularization term and the $H^{-1}$ norm as a fidelity term. Using the properties of the fractional Laplacian operator, the $L^p$ norm is employed with an adjustable parameter p to enhance the operator’s ability to restore fine details in various types of images. The replacement of the conventional $L^2$ norm with the $H^{-1}$ norm enables better preservation of global structures in denoising and restoration tasks. This paper introduces a diffusion partial differential equation by adding an intermediate term and provides a theoretical proof of the existence and uniqueness of its solution in Sobolev spaces. Furthermore, it demonstrates that the solution converges to the minimizer of the energy functional as time approaches infinity. Numerical experiments that compare the proposed method with traditional and deep learning models validate its effectiveness in image inpainting tasks. Full article

This paper presents a high-order structure-preserving difference scheme for the nonlinear space fractional sine-Gordon equation with damping, employing the triangular scalar auxiliary variable approach. The original equation is reformulated into an equivalent system that satisfies a modified energy conservation or dissipation law, significantly reducing the computational complexity of nonlinear terms. Temporal discretization is achieved using a second-order difference method, while spatial discretization utilizes a simple and easily implementable discrete approximation for the fractional Laplacian operator. The boundedness and convergence of the proposed numerical scheme under the maximum norm are rigorously analyzed, demonstrating its adherence to discrete energy conservation or dissipation laws. Numerical experiments validate the scheme’s effectiveness, structure-preserving properties, and capability for long-time simulations for both one- and two-dimensional problems. Additionally, the impact of the parameter $\epsilon$ on error dynamics is investigated. Full article

We introduce a novel family of compactly supported basis functions, termed Legendre Delta-Shaped Functions (LDSFs), constructed using the eigenfunctions of the Legendre differential equation. We begin by proving that LDSFs form a basis for a Haar space. We then demonstrate that interpolation using classical Legendre polynomials is a special case of interpolation with the proposed Legendre Delta-Shaped Basis Functions (LDSBFs). To illustrate the potential of LDSBFs, we apply the corresponding series to approximate a rectangular pulse. The results reveal that Gibbs oscillations decay rapidly, resulting in significantly improved accuracy across smooth regions. This example underscores the effectiveness and novelty of our approach. Furthermore, LDSBFs are employed within the collocation framework to solve Poisson-type equations and systems of nonlinear differential equations arising in energy transfer problems. We also derive new error bounds for interpolation polynomials in a special case, expressed in both the discrete ${(L}_2)$ norm and the Sobolev $\left(H_p\right)$ norm. To validate the proposed method, we compare our results with those obtained using the Legendre pseudospectral method. Numerical experiments confirm that our approach is accurate, efficient, and highly competitive with existing techniques. Full article

This paper proposes a novel sparse twin parametric insensitive support vector regression (STPISVR) model, designed to enhance sparsity and improve generalization performance. Similar to twin parametric insensitive support vector regression (TPISVR), STPISVR constructs a pair of nonparallel parametric insensitive bound functions to indirectly determine the regression function. The optimization problems are reformulated as two sparse linear programming problems (LPPs), rather than traditional quadratic programming problems (QPPs). The two LPPs are originally derived from initial L1-norm regularization terms imposed on their respective dual variables, which are simplified to constants via the Karush–Kuhn–Tucker (KKT) conditions and consequently disappear. This simplification reduces model complexity, while the constraints constructed through the KKT conditions— particularly their geometric properties—effectively ensure sparsity. Moreover, a two-stage hybrid tuning strategy—combining grid search for coarse parameter space exploration and Bayesian optimization for fine-grained convergence—is proposed to precisely select the optimal parameters, reducing tuning time and improving accuracy compared to a singlemethod strategy. Experimental results on synthetic and benchmark datasets demonstrate that STPISVR significantly reduces the number of support vectors (SVs), thereby improving prediction speed and achieving a favorable trade-off among prediction accuracy, sparsity, and computational efficiency. Overall, STPISVR enhances generalization ability, promotes sparsity, and improves prediction efficiency, making it a competitive tool for regression tasks, especially in handling complex data structures. Full article

Photovoltaic (PV) models are hard to optimize due to their intrinsic complexity and changing operation conditions. Root mean square error (RMSE) is often given precedence in classic single-objective optimization methods, limiting them to address the intricate nature of PV model calibration. To bypass these limitations, this research proposes a novel multi-objective optimization framework balancing accuracy and robustness by considering both maximum error and the L2 norm as significant objective functions. Along with that, we introduce the Random Search Around Bests (RSAB) algorithm, which is a parameterless metaheuristic designed to be effective at exploring the solution space. The primary contributions of this work are as follows: (1) an extensive performance evaluation of the proposed framework; (2) an adaptable function to adjust dynamically the trade-off between robustness and error minimization; and (3) the elimination of manual tuning of the RSAB parameters. Rigorous testing across three PV models demonstrates RSAB’s superiority over 17 state-of-the-art algorithms. By overcoming significant issues such as premature convergence and local minima entrapment, the proposed procedure provides practitioners with a reliable tool to optimize PV systems. Hence, this research supports the overarching goals of sustainable energy technology advancements by offering an organized and flexible solution enhancing the accuracy and efficiency of PV modeling, furthering research in renewable energy. Full article

This article examines the intersecting forces of gender, class, and education in early twentieth-century Britain through a feminist reading of Pip Williams’ historical novel The Bookbinder of Jericho. Centering on the fictional character Peggy Jones—a working-class young woman employed in the Oxford University Press bindery—the study explores how women’s intellectual ambitions were constrained by economic hardship, institutional gatekeeping, and patriarchal social norms. By integrating close literary analysis with historical research on women bookbinders, educational reform, and the impact of World War I, the paper reveals how the novel functions as both a narrative of personal development and a broader critique of systemic exclusion. Drawing on the genre of the female Bildungsroman, the article argues that Peggy’s journey—from bindery worker to aspiring scholar—mirrors the real struggles of working-class women who sought education and recognition in a male-dominated society. It also highlights the significance of female solidarity, especially among those who served as volunteers, caregivers, and community organizers during wartime. Through the symbolic geography of Oxford and its working-class district of Jericho, the novel foregrounds the spatial and social divides that shaped women’s lives and labor. Ultimately, this study shows how The Bookbinder of Jericho offers not only a fictional portrait of one woman’s aspirations but also a feminist intervention that recovers and reinterprets the overlooked histories of British women workers. The novel becomes a literary space for reclaiming agency, articulating resistance, and criticizing the gendered boundaries of knowledge, work, and belonging. Full article

A lightweight YOLOv8 variant, CIMB-YOLOv8, is proposed to address challenges in remote sensing object detection, such as complex backgrounds and multi-scale targets. The method enhances detection accuracy while reducing computational costs through two key innovations: Contextual Multi-branch Fusion: Integrates a space-to-depth multi-branch pyramid (SMP) to capture rich contextual features, improving small target detection by 1.2% on DIOR; Lightweight Architecture: Employs Lightweight GroupNorm Detail-enhance Detection (LGDD) with shared convolution, reducing parameters by 14% compared to YOLOv8n. Extensive experiments on DIOR, DOTA, and NWPU VHR-10 datasets demonstrate the model’s superiority, achieving 68.1% mAP on DOTA (+0.7% over YOLOv8n) and 82.9% mAP on NWPU VHR-10 (+1.7%). The model runs at 118.7 FPS on NVIDIA 3090, making it well-suited for real-time applications on resource-constrained devices. Results highlight its practical value for remote sensing scenarios requiring high-precision and lightweight detection. Full article

This paper provides a new proof of the operator norm identity $\parallel Q\parallel = \parallel I-Q\parallel$, where Q is a bounded idempotent operator on a complex Hilbert space, and I is the identity operator. We also derive explicit lower and upper bounds for the distance from an arbitrary idempotent operator to the set of orthogonal projections. Our approach simplifies existing proofs. Full article