ABSTRACT In this paper, we study the ground states of spin‐2 Bose‐Einstein condensates in with spin‐independent interaction constant and spin‐exchange interaction constant . Two conserved quantities are involved: the number of atoms and the total magnetization . We mainly investigate the existence and strong instability of ground‐state standing waves in the case of no external potential. The bounded Palais‐Smale sequence satisfying the Pohozaev identity in the limit sense is obtained by a minimax theorem. Applying the technique of mass‐redistribution, we further determine the signs of corresponding Lagrange multipliers, which allows the strong convergence of the approximate critical point sequence in . Our method presents an alternative proof for excluding semi‐trivial solutions already available in some literature.

In this paper, we construct a novel iterative scheme for approximating fixed points of generalized [Formula: see text]-nonexpansive mappings in the setting of a real Banach space. The proposed scheme not only generalizes but also unifies and extends several well-known fixed point iterative processes available in the literature. We establish both weak and strong convergence results under appropriate conditions. Furthermore, a comparative analysis of the rate of convergence is carried out using a carefully chosen numerical example, with the outcomes demonstrated through both tabular and graphical illustrations.In addition to convergence properties, we derive a data dependence result, offering insights into the stability of the proposed scheme with respect to perturbations in the underlying mapping. We further prove that the scheme satisfies [Formula: see text]-stability and almost [Formula: see text]-stability criteria, thereby enhancing its robustness in practical applications. To demonstrate the applicability of our results, we provide significant application of the analysis to a SEIR epidemic model governed by a Caputo-type fractional differential equation, showcasing the utility of the proposed method in the context of real-world dynamical systems. Our findings contribute to the advancement of fixed point theory and its applications in mathematical modeling, offering a flexible and powerful tool for analyzing complex nonlinear problems.

Strong and weak convergence orders of numerical methods for SDEs driven by time-changed Lévy noise

Abstract We introduce and investigate the asymptotic behaviour of the trajectories of a second order dynamical system with Tikhonov regularization for solving a monotone equation with single valued, monotone and continuous operator acting on a real Hilbert space. We consider a vanishing damping which is correlated with the Tikhonov parameter and which is in line with recent developments in the literature on this topic. A correction term which involves the time derivative of the operator along the trajectory is also involved in the system and makes the link with Newton and Levenberg-Marquardt type methods. We obtain strong convergence of the trajectory to the minimal norm solution and fast convergence rates for the velocity and a quantity involving the operator along the trajectory. The rates are very close to the known fast convergence results for systems without Tikhonov regularization, the novelty with respect to this is that we also obtain strong convergence of the trajectory to the minimal norm solution. As an application we introduce a primal-dual dynamical system for solving linearly constrained convex optimization problems, where the strong convergence of the trajectories is highlighted together with fast convergence rates for the feasibility measure and function values.

Abstract This paper investigates the approximation of fixed points for quasi-nonexpansive mappings in Banach spaces using the general Picard-Mann (GPM) algorithm. Under mild conditions such as the demiclosedness of $$I - \digamma $$ I - ϝ at zero and the Opial property, we derive weak and strong convergence theorems for the iterative sequences generated by the GPM method. We establish several stability results for the GPM scheme, including summably almost stability property for quasi-contractive mappings. The theoretical findings are applied to the classical relaxation method for solving systems of linear inequalities, demonstrating the practical relevance of our approach. Numerical examples in $$\mathbb {R}^4$$ R 4 and $$\ell ^2$$ ℓ 2 are provided to illustrate the efficiency and convergence behavior of the proposed algorithm. The results presented herein extend and complement existing work in fixed point theory and iterative approximation methods.

Quantum Bernoulli noises (QBN) are the family of annihilation and creation operators acting on Bernoulli functionals, which satisfy a canonical anti-communication relation (CAR) in equal-time. This paper introduces a new type of fermionic open quantum random walk by formulating it with QBN on single-excitation subspaces. Within this framework, we establish a rigorous criterion for irreducibility, which requires the conjunction of strong graph connectivity, algebraic irreducibility of the transition operators, and non-vanishing transition amplitudes. For finite irreducible walks, we prove convergence to a unique invariant state and establish strong convergence under the additional condition of aperiodicity. The theoretical framework is validated through a comprehensive two-node example, which illustrates both the convergence behavior and the intrinsic quantum features of the model.

We investigate the asymptotic behaviour of fast rotating incompressible fluids with vanishing viscosity, in a three dimensional domain with topography including the case of land area. Assuming the initial data is well-prepared, we prove a convergence theorem of the velocity fields to a two-dimensional vector field solving a linear, damped ordinary differential equation. The proof is based on a weak-strong uniqueness argument, combined with an abstract result implying that the weak convergence of a family of weak solutions to the Navier-Stokes-Coriolis system can be translated into a form of uniform-in-time convergence. This argument yields strong convergence of the velocity fields, without a precise rate though.

ABSTRACT Background Obsessive – compulsive disorder (OCD) is a chronic condition that disrupts functioning and quality of life. The Yale – Brown Obsessive – Compulsive Scale (Y-BOCS; clinician-administered) and its Self-Report version (Y-BOCS-SR) are recognised gold standards for assessing OCD severity. Previous Turkish adaptations have been partial, limited by small samples or by validating only one format. This study provides the first comprehensive Turkish validation of both instruments. Methods The sample comprised 950 adults: 158 with DSM-5-TR – diagnosed OCD and 792 non-clinical adults. Translation followed a multi-phase process involving clinicians and a linguistics expert. Participants completed the Turkish Y-BOCS, Y-BOCS-SR, and additional OCD measures (OCI-R, VOCI, PI-R). Results The clinician-administered Turkish Y-BOCS demonstrated excellent reliability (α = 0.91; ICC = 0.88) and strong psychometric validity, including a stable two-factor structure distinguishing obsessions and compulsions. The self-report version showed comparably high internal consistency (α = 0.90) and strong convergence with the clinician-administered scale. Convergent and criterion validity were robust, with high sensitivity and specificity. Cross-cultural comparisons further supported consistency and generalisability. Conclusions This study establishes both the clinician-administered and self-report Turkish Y-BOCS as reliable, valid, and culturally sensitive tools for assessing OCD severity in clinical and research contexts.

In recent years, significant progress has been made in understanding the theoretical properties of estimators in deep neural networks. However, most of the existing research is limited to scenarios involving bounded loss functions or bounded input/output data. To the best of our knowledge, there is currently no work that addresses the strong convergence of DNN estimators in general settings. In this paper, we investigate the rate of strong convergence for generalisation bounds and excess risk in learning ψ-weakly dependent processes and strong mixing processes, where the loss function and input/output are not necessarily bounded. Under some appropriate assumptions, the asymptotic learning rate approximates o ( n − μ + 1 2 μ + 3 ) with μ ≥ 0 for ψ-weakly dependent processes and o ( n − 1 / τ ) with 2 $ ]]> τ > 2 for strong mixing processes. We also provide some simple simulation results to support our findings.

A sudden surge in urbanization and population has escalated challenges in cities in waste generation, making efficient route optimization for waste management a critical necessity. This study proposes a hybrid metaheuristic framework, Locally Optimized Discrete Cuckoo Search (LO-DCS), for an effective route optimization in urban waste management. The proposed algorithm adapts the classical Cuckoo Search algorithm to discrete routing problems by integrating permutation-based random walk, 2-opt local optimization and K-means clustering. The input data is obtained from waste bin coordinates which were extracted using Google Earth Engine and georeferenced satellite imagery within a predefined region of interest. The proposed framework was implemented on real-world urban datasets from Bengaluru city using multiple performance indicators, including travel distance, fuel consumption, carbon emission and operational time. Extensive experiments involving 30 independent runs were performed to assess stability and robustness. Comparative analysis with Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Discrete Spider Monkey Optimization (DSMO) and Quantum-based Avian Navigation algorithm (QANA) demonstrates the competitive performance of LO-DCS across all the bin clusters. Statistical significance test was used to validate the results using Wilcoxon and Friedman tests with Holm correction. Furthermore, optimality gap analysis using exact solvers confirms that LO-DCS produces near-optimal solutions for moderate-sized bin-cluster instances. The experimental results show that LO-DCS achieves an average improvement of approximately 85% across the clusters for all the key performance indicators (distance, fuel consumption, CO2 emission and travel time). When compared with the baseline methods, it achieves an improvement of 78% approximately with a strong convergence behaviour. The implemented approach provides a scalable, data-driven decision-support tool for sustainable and cost-effective urban waste management. The municipal authorities and researchers can gain valuable insights from the findings toward environmentally responsible infrastructure planning.

In this paper, we study bilevel equilibrium (with a pseudomonotone bifunction) and inclusion problems in the framework of Hilbert spaces. We propose an inertial subgradient extragradient algorithm with self-adaptive step size for finding common solutions to the aforementioned problems. Unlike several existing works on inclusion problem in the literature, our under lying operator is monotone and Lipschitz continuous. Also, we also employ the use of inertial technique to accelerate the rate of strong convergence of the sequences generated by our proposed method. Moreover, the implementation of our proposed method does not require prior knowledge of the Lipschitz constant of the monotone operator. Furthermore, we give some numerical examples to illustrate the efficiency of our proposed algorithm.

ABSTRACT This study examines how Yi students utilise Chinese (Mandarin and Sichuan dialect), Yi, and English across both formal and informal contexts in two bilingual education models in Liangshan, China. Although policies stipulate the use of Yi language (Model One) or Chinese (Model Two) as the instructional language, comprehensive analyses, including observations, interviews, language maps, and linguistic landscape, reveal a strong convergence in language practices across both models. These methods allow the study to trace how convergence materialises across formal and informal practices and public semiotic environments. Yi literacy was generally taught for examination purposes rather than everyday communication. Outside the classroom, students spoke Yi with family and close peers, but they used Chinese in public places and for written expressions, including note-taking and social messaging. The findings highlight a policy-practice gap and suggest the need for pedagogical reforms that enhance the functional role of Yi scripts in students’ daily lives.

Singular limits for the following indirect signalling chemotaxis system are investigated. More precisely, we study parabolic-elliptic simplification, or PES, with fixed up to the critical dimension , and indirect-direct simplification, or IDS, up to the critical dimension . These are relevant in biological situations where the signalling process is on a much faster time scale compared to the species diffusion and all interactions. Showing singular limits in critical dimensions is challenging. To deal with the PES, we carefully combine the entropy function, an Adam-type inequality, the regularisation of slow evolution, and an energy equation method to obtain strong convergence in representative spaces. For the IDS, a bootstrap argument concerning the -energy function is devised, which allows us to obtain suitable uniform bounds for the singular limits. Moreover, in both scenarios, we also present the convergence rates, where the effect of the initial layer and the convergence to the critical manifold are also revealed.

Abstract We are concerned with the existence of solution of the problem where $$\Delta ^H_pu=\hbox {div }(a(\nabla u))$$ Δ p H u = div ( a ( ∇ u ) ) , with $$a(\xi )=H^{p-1}(\xi )\nabla H(\xi ),\, \xi \in \mathbb {R}^N,$$ a ( ξ ) = H p - 1 ( ξ ) ∇ H ( ξ ) , ξ ∈ R N , $$N\geqslant 3,$$ N ⩾ 3 , is the anisotropic p -Laplacian with $$1<p<N$$ 1 < p < N , $$\lambda >0$$ λ > 0 is a parameter, and $$p< q<p^*=pN/(N-p)$$ p < q < p ∗ = p N / ( N - p ) . Further, $$\Omega $$ Ω is a $$C^1$$ C 1 bounded domain inside a convex open cone. To succeed with a variational approach, where the strong convergence of a bounded (PS) subsequence needs to be proved, one has to deal with anisotropic norms in the absence of a Tartar’s type inequality, unlike the isotropic p -Laplace case. This is overcome by proving the a.e. convergence of its gradients. Furthermore, the solution of ( P ) is shown to belong to $$C^{1,\alpha }(\Omega )$$ C 1 , α ( Ω ) from classical elliptic regularity theory, and is positive from a Harnack inequality, since any solution of ( P ) is bounded. This in turn is a consequence of a result we prove which assures that any $$W^{1,p}$$ W 1 , p -solution of critical Neumann problems with the anisotropic p -Laplacian operator on bounded Lipschitz domains in $$\mathbb {R}^N$$ R N $$(N\geqslant 3)$$ ( N ⩾ 3 ) is bounded.

In this research, we introduce an improved pseudomonotone subgradient extragradient algorithm for finding common solutions of equilibrium and fixed point problems in real Hilbert spaces. We obtain the strong convergence results of the proposed method under some mild and suitable assumptions on the control parameters. Unlike many existing methods that rely on contraction, and Mann-like techniques to obtain strong convergence, our method employs the typical Mann iteration technique which does requires complex computations. Furthermore, our method incorporates a relaxed two-inertial technique which enhances its speed of convergence. Additionally, we demonstrate the applicability of our findings to variational inequality problems, and image recovery problems. Finally, we present some numerical experiments to validate our theoretical results and show the superiority of our method over some well known results in the literature. The obtained results in this paper improve, extend and unify many existing results in this research direction.

The integration of artificial intelligence (AI) into history education remains underexplored in peripheral regions of Indonesia, where infrastructure limitations and concerns about cultural representation intersect with the imperative to innovate pedagogy. This mixed-methods study examined teachers' and students' perceptions of AI integration in history learning in Palangka Raya, Central Kalimantan. Participants included 220 Grade XI students and six history teachers from six public senior high schools, selected through purposive sampling for school selection and maximum variation sampling to ensure diversity in socioeconomic background, technological access, and geographic location. Data collection involved a validated 30-item questionnaire measuring five perceptual dimensions (AI understanding, attitudes, readiness, expectations, and barriers), along with 12 in-depth student interviews and six semi-structured teacher interviews. Quantitative analysis revealed moderate AI understanding (M = 3.19), strongly positive attitudes (M = 4.09), moderate readiness (M = 3.28), high expectations (M = 4.10), and awareness of implementation barriers (M = 2.75). Thematic analysis identified strong convergence on AI's complementary role rather than a replacement for human educators. Both groups emphasized concerns about cultural representation and infrastructure constraints. The findings recommend context-responsive AI content co-designed with local communities and equity-focused infrastructure investment to support inclusive digital pedagogy in marginalized regions.

The problem of bandwidth selection in nonparametric classification is considered when a large number of class variables may be missing, but not necessarily missing at random. This setup is generally acknowledged to be more challenging than the so-called missing at random setup. Our proposed approach starts by constructing a family of kernel-based classifiers where the members of the family are indexed by the kernel bandwidth h as well as the nonignorability parameter of the selection probability mechanism. Next, a search is performed to find the member of a finite cover of this family that has the smallest empirical misclassification error. To assess the performance of the resulting classifiers, we derive exponential performance bounds on the deviations of their errors from that of the theoretically optimal classifier. These bounds are then used to study strong convergence properties of the proposed classifiers. Our theoretical findings are further confirmed by numerical studies.

Abstract We propose a comprehensive framework for solving constrained variational inequalities via various classes of evolution equations displaying multi-scale aspects. In an infinite-dimensional Hilbertian framework, the class of dynamical systems we propose combine Tikhonov regularization and exterior penalization terms in order to induce strong convergence of trajectories to least norm solutions in the constrained domain. Our construction thus unifies the literature on regularization methods and penalty-based dynamical systems. An extension to a full splitting formulation of the constrained domain is also provided, with associated weak convergence results involving the Attouch-Czarnecki condition.

We develop a fully discrete, semi-implicit mixed finite element method for approximating solutions to a class of fourth-order stochastic partial differential equations (SPDEs) with non-globally Lipschitz and non-monotone nonlinearities, perturbed by spatially smooth multiplicative Gaussian noise. The proposed scheme is applicable to a range of physically relevant nonlinear models, including the stochastic Landau--Lifshitz--Baryakhtar (sLLBar) equation, the stochastic convective Cahn--Hilliard equation with mass source, and the stochastic regularised Landau--Lifshitz--Bloch (sLLB) equation, among others. To overcome the difficulties posed by the interplay between the nonlinearities and the stochastic forcing, we adopt a `truncate-then-discretise' strategy: the nonlinear term is first truncated before discretising the resulting modified problem. We show that the strong solution to the truncated system converges in probability to that of the original problem. A fully discrete numerical scheme is then proposed for the truncated problem. Assuming initial data in $\mathbb{H}^2$, we utilise parabolic smoothing estimates and the temporal H\"older continuity of the solution to establish both convergence in probability and strong convergence (with quantitative rates) for the two fields used in the mixed formulation. Numerical simulations are provided to support the theoretical results.

In this paper, we propose a viscosity-type CQ algorithm, solving the split equality problem by using two adaptive step sizes and combining the alternated inertial technique in Hilbert space. The strong convergence of the sequences generated by the proposed algorithm is proved under some mild conditions. Finally, numerical experiments are conducted to verify the effectiveness and superiority of the method. The research results are innovative and provide new expansion and supplement to the relevant research.