Search Results (94)

The importance of sequence spaces has increased with the emergence of various new convergence methods such as statistical convergence. On the other hand, partial metric spaces hold an important place in computer science, data science, and convergence analysis because they contain points whose distance from themselves is non-zero. For these reasons, in the present paper, we generalize the concept of statistical convergence, previously defined for cone metric spaces, to partial cone metric spaces, defining statistical convergence of order $\alpha$ and $\lambda$-statistical convergence of order $\alpha$ and demonstrating their relationships. Furthermore, we define the statistical Cauchy sequences of order $\alpha$ and the $\lambda$-statistical boundedness of order $\alpha$ and examine some inclusion theorems. Additionally, in partial cone metric spaces, we show that a non-convergent sequence is statistically convergent of order $\alpha$. Full article

Although DNA-based data storage theoretically provides an information density of 2 bits per nucleotide, biochemical constraints transform sequence design into a high-dimensional constrained combinatorial optimization problem. The high computational cost and low encoding efficiency of conventional rule-based approaches make metaheuristic methods an effective alternative. This study proposes the TC-HUR hybrid algorithm to simultaneously optimize information density and conflicting biophysical constraints, including homopolymer (HP) length, GC content, melting temperature ($T_m$), and reverse-complement (RC) similarity. The method escapes local optima using Cauchy jump-enhanced Hunger Games Search (HGS), performs high-precision exploitation via Runge–Kutta (RUN) operators, and refines constraint violations at the nucleotide level through an adaptive intensive mutation mechanism. The algorithm is evaluated on a complex dataset of 1853 nucleotides under different noise regimes. TC-HUR outperforms RUN by 2.5% and HGS by 16.7% in average fitness. While maintaining homopolymer length near the ideal threshold, it reduces reverse-complement similarity to 19.10%, ensuring high sequence diversity. Under high-noise conditions, TC-HUR achieves a normalized edit distance of 0.1290, reducing insertion–deletion (indel) errors by approximately 14%. The results demonstrate that the proposed model effectively generates biophysically synthesizable and noise-resilient DNA codes. Full article

Radar image extrapolation serves as a fundamental methodology in meteorological forecasting, facilitating precise short-term weather prediction through spatiotemporal sequence analysis. Conventional approaches remain constrained by progressive image degradation and artifacts, substantially limiting operational forecasting reliability. This research introduces E-HEOA—an enhanced deep learning architecture with integrated hyperparameter optimization. Our framework incorporates two fundamental innovations: (a) a hybrid metaheuristic optimizer merging a Gaussian-mutated ESOA and Cauchy-mutated HEOA for autonomous learning rate and dropout optimization and (b) embedded ConvLSTM2D modules for enhanced spatiotemporal feature preservation. Experimental validation on 170,000 radar echo samples demonstrates superior performance, demonstrating considerable enhancement in almost all aspects relative to the baseline model while establishing new state-of-the-art benchmarks in prediction fidelity, convergence efficiency, and structural similarity metrics. Full article

This Editorial introduces the Symmetry Special Issue “Theory and Applications of Special Functions II” and summarizes the nine contributions collected therein. The papers span the analytic continuation of multivariate hypergeometric functions; stability theory for differential equations via integral transforms; numerical schemes for multi-space fractional partial differential equations based on nonstandard finite differences and orthogonal polynomials; applications of the Lambert W function to viscoelastic creep modeling; algebraic constructions of new Hermite-type polynomial families via the monomiality principle; higher-level generalizations of poly-Cauchy numbers; Bell-polynomial expansions for Laplace transforms of higher-order nested functions; and two complementary studies on the physical implementation and algebraic description of Gaussian quantum states. Beyond the contributions of the Special Issue, we highlight methodological connections—continued fractions and complex analysis, transform techniques, special polynomials, and combinatorial sequences—and emphasize the unifying role of symmetry across mathematical structures and applications. Full article

Some properties on large contractions in metric spaces are proven. In particular, such contractions are proven to be asymptotically regular. In addition, if the metric space is complete, then the sequences that they generate are bounded, Cauchy, and convergent to a unique fixed point. Also, cyclic large contractions are an area of focus. It is proven that, if subsets of the cyclic disposal are nonempty closed and they intersect, all the sequences are bounded and Cauchy, and they converge to a unique fixed point located in the intersection of such subsets if the metric space is complete. If the subsets have a pair-wise empty intersection, then the boundedness of such sequences is proven without the need to assume the boundedness of the subsets in the cyclic disposal. The convergence of the sequences to a unique limit cycle of best proximity points, with one per subset in the cyclic disposal, is proven provided that the metric space is complete and that one of such subsets is boundedly compact with a singleton best proximity set. For that property to hold, it is not assumed that the remaining best proximity points are necessarily singletons. It has also been proven that all the subsequences contained within each of the subsets are Cauchy and they converge to a unique best proximity point, even if the corresponding best proximity sets is not a singleton. Furthermore, the hypothesis that one of the best proximity sets between adjacent subsets is a singleton can be weakened for any particular cyclic large contraction. Later on, eventual perturbations of the cyclic large self-mappings in normed spaces are discussed. If the norm of the perturbation additive operator is small enough, it is proven that the perturbed cyclic self-mapping maintains the property of being a cyclic large contraction associated with the unperturbed nominal cyclic large contraction. The maximum upper-bound of the perturbed operator ensures that such a property is given in an explicit manner. Full article

After the introduction of the relation-theoretic contraction principle, the branch of metric fixed-point theory has attracted much attention in this direction, and various fixed-point results have been proven in the framework of relational metric space via different approaches. The aim of this article is to establish some fixed-point outcomes in the framework of relational metric space verifying a generalized nonlinear contraction utilizing three test functions $\mathrm{\Phi }$, $\mathrm{\Psi }$ and $\mathrm{\Theta }$ satisfying the appropriate characteristics. The findings obtained herein expand, sharpen, improve, modify and unify a few well-known findings. To demonstrate the utility of our outcomes, several examples are furnished. We utilized our outcomes to investigate a unique solution of second-order ordinary differential equations prescribed with specific boundary conditions. Full article

In intuitionistic fuzzy 2-normed spaces, there are numerous symmetries in the topological structures and mapping theorems. In this work, we present the concept of an intuitionistic fuzzy 2-normed space(IF2NS) and demonstrate its structural properties using illustrative examples. This approach unifies and broadens the scope of both classical 2-normed spaces and intuitionistic fuzzy normed spaces when specific conditions are met. We introduce the idea of fuzzy open balls and explore the convergence of sequences with respect to the topology derived from the intuitionistic fuzzy 2-norm. In addition, we define left and right N-Cauchy sequences relative to the topologies ${\tau }_N$ and ${\tau }_{N^{-1}}$ and analyze their convergence characteristics. Special attention is given to the inherent symmetry of the 2-norm, where the magnitude of a pair of vectors remains invariant under exchange of arguments, and to the balanced interaction between membership and non-membership functions in the intuitionistic fuzzy setting. This intrinsic symmetry is further reflected in the proofs of the open mapping and closed graph theorems, which naturally preserve the symmetric structure of the underlying space The paper culminates with the formulation and proof of the open mapping theorem that can be considered for its symmetric properties and the closed graph theorem in the context of IF2NS, thereby generalizing essential theorems of functional analysis to this fuzzy setting. Full article

We present a unified framework for fuzzy statistical convergence of Grünwald–Letnikov (GL) fractional differences in Bag–Samanta fuzzy normed linear spaces, addressing memory effects and nonlocality inherent to fractional-order models. Theoretically, we establish the uniqueness, linearity, and invariance of fuzzy statistical limits and prove a Cauchy characterization: fuzzy statistical convergence implies fuzzy statistical Cauchyness, while the converse holds in fuzzy-complete spaces (and in the completion, otherwise). We further develop an inclusion theory linking fuzzy strong Cesàro summability—including weighted means—to fuzzy statistical convergence. Via the discrete Q-operator, all statements transfer verbatim between nabla-left and delta-right GL forms, clarifying the binomial GL↔discrete Riemann–Liouville correspondence. Beyond structure, we propose density-based residual diagnostics for GL discretizations of fractional initial-value problems: when GL residuals are fuzzy statistically negligible, trajectories exhibit Ulam–Hyers-type robustness in the fuzzy topology. We also formulate a fuzzy Korovkin-type approximation principle under GL smoothing: Cesàro control on the test set $\{1,x,x^2\}$ propagates to arbitrary targets, yielding fuzzy statistical convergence for positive-operator sequences. Worked examples and an engineering-style case study (thermal balance with memory and bursty disturbances) illustrate how the diagnostics certify robustness of GL numerical schemes under sparse spikes and imprecise data. Full article

Padé approximants are computational tools customarily employed for resumming divergent Stieltjes series. However, they become ineffective or even fail when applied to Stieltjes series whose moments do not satisfy the Carleman condition. Differently from Padé, Levin-type transformations incorporate important structural information on the converging factors of a typical Stieltjes series. For example, the computational superiority of Weniger’s $\delta$-transformation over Wynn’s epsilon algorithm is ultimately based on the fact that Stieltjes series converging factors can always be represented as inverse factorial series. In the present paper, the converging factors of an important class of superfactorially divergent Stieltjes series are investigated via an algorithm developed one year ago from the first-order difference equation satisfied by the Stieltjes series converging factors. Our analysis includes the analytical derivation of the inverse factorial representation of the moment ratio sequence of the series under investigation, and demonstrates the numerical effectiveness of our algorithm, together with its implementation ease. Moreover, a new perspective on the converging factor representation problem is also proposed by reducing the recurrence relation to a linear Cauchy problem whose explicit solution is provided via Faà di Bruno’s formula and Bell’s polynomials. Full article

Solar greenhouses are an important component of modern facility agriculture, and the dynamic changes in their internal environment directly affect crop growth and yield. Among these factors, crop transpiration releases water vapor through transpiration, directly altering the indoor humidity balance and forming a dynamic coupling with factors such as temperature and light. The environment of solar greenhouses exhibits highly nonlinear and multivariate coupling characteristics, leading to insufficient prediction accuracy in existing models. However, accurate predictions are crucial for regulating crop growth and yield. However, current mainstream greenhouse environmental prediction models still have obvious limitations when dealing with such complexity: traditional machine learning models and single-variable-driven models have issues such as insufficient accuracy (average MAE is 15–20% higher than in this study) and weak adaptability to nonlinear environmental changes in multi-environmental factor coupling predictions, making it difficult to meet the needs of precision farming. A review of relevant research over the past five years shows that while LSTM-based models perform well in time series prediction, they ignore the spatial correlations between environmental factors. Models incorporating attention mechanisms can capture key variables but suffer from high computational costs. To address these issues, this study proposes a prediction model based on multi-strategy optimization and gradient-boosting (GBDT) algorithms. By introducing a multi-scale feature fusion module, it addresses the accuracy issues in multi-factor coupling prediction. Additionally, it employs a lightweight network design to balance prediction performance and computational efficiency, filling the gap in existing research applications under complex greenhouse environments. The model optimizes data preprocessing and model parameters through Sobol sequence initialization, adaptive t-distribution perturbation strategies, and Gaussian–Cauchy mixture mutation strategies and combines CatBoost for modeling to enhance prediction accuracy. Experimental results show that the MSCSO–CatBoost model performs excellently in temperature prediction, with the mean absolute error (MAE) and root mean square error (RMSE) reduced by 22.5% (2.34 °C) and 24.4% (3.12 °C), respectively, and the coefficient of determination (R²) improved to 0.91, significantly outperforming traditional regression methods and combinations of other optimization algorithms. Additionally, the model demonstrates good generalization capability in predicting multiple environmental variables such as temperature, humidity, and light intensity, adapting to environmental fluctuations under different climatic conditions. This study confirms that combining multi-strategy optimization with gradient-boosting algorithms can significantly improve the prediction accuracy of solar greenhouse environments, providing reliable support for precision agricultural management. Future research could further explore the model’s adaptive optimization in complex climatic regions. Full article

Inspired by the work of Gregori et al. and guided by some open direction, we propose the concepts of Cauchy sequence and convergent sequence in a fuzzy partial metric space by the residuum operator associated to a continuous t-norm. Based on these notions, we introduce the concepts of two kinds of fuzzy $\eta$-contractive mappings in fuzzy partial metric spaces and present related fixed point theorems. Full article

This paper introduces and analyzes the concept of $\lambda$-statistical convergence in fuzzy paranormed spaces, demonstrating its relevance to supply chain inventory management under demand shocks. We establish key relationships between generalized convergence methods and fuzzy convex analysis, showing how these results extend classical summability theory to uncertain demand environments. By exploring $\lambda$-statistical Cauchy sequences and $(V,\lambda )$-summability in fuzzy paranormed spaces, we provide new insights applicable to adaptive inventory optimization and decision-making in supply chains. Our findings bridge theoretical aspects of fuzzy convexity with practical convergence tools, advancing the robust modeling of demand uncertainty. Full article

The problem of allocating multiple UAV tasks is a complex combinatorial optimization challenge, involving various constraints. This paper presents an autonomous multi-UAV cooperative task allocation method based on an improved Great Wall Construction Algorithm. A model integrating battlefield environmental factors, 3D terrain data, and threat assessments is developed for optimized task allocation and trajectory planning. The algorithm is enhanced using a good point set initialization strategy, Gaussian distribution estimation, and a Cauchy reorganization variant. The simulation results show that replacing straight-line distances with actual flight distances leads to more rational mission sequences, improving combat effectiveness under realistic terrain and threat conditions. The enhanced algorithm demonstrates superior accuracy and faster convergence. Full article

The concept of fuzzy metric space introduced by Kramosil and Michalek was later slightly modified by George and Veeramani who imposed three additional restrictions on it. A significant difference between these two concepts of fuzzy metrics is that fuzzy metric spaces in the sense of George and Veeramani do not admit completion, in general. This paper is devoted to go into detail on completable fuzzy metric spaces by means of the study of the impact on the completion of each one of the restrictions imposed by George and Veeramani in their definition of fuzzy metric. In this direction, we characterize those completable fuzzy metric spaces, in which just one of the three restrictions imposed by George and Veeramani is required. Various examples illustrate and justify the main results. Full article

This paper explores the concepts of $\mathfrak{J}$-lacunary statistical limit points, $\mathfrak{J}$-lacunary statistical cluster points, and $\mathfrak{J}$-lacunary statistical Cauchy multiset sequences. Building upon previous work in the field, we investigate the relationships between $\mathfrak{J}$-lacunary statistical convergence and ${\mathfrak{J}}^{*}$-lacunary statistical convergence in multiset sequences. The findings contribute to a deeper understanding of the convergence behaviour of multiset sequences and provide new insights into the application of ideal convergence in this context. Full article