Elliptic Curve Cryptosystems (ECC) have emerged as a powerful alternative to traditional public-key cryptosystems, offering equivalent security with significantly smaller key sizes. The efficiency of ECC, in terms of minimizing encryption time and enhancing computational performance, is strongly influenced by the number of point addition and doubling computations required in elliptic curve arithmetic. In this context, the study of arithmetic of points in elliptic curves plays a crucial role. This study emphasizes computations in projective coordinates of points on elliptic curves defined over the <img src=image/13443798_05.gif>-adic field <img src=image/13443798_06.gif>. A comparative study shows that arithmetic in projective coordinates reduces the number of operations required, thereby enhancing the efficiency relative to the affine coordinate system. The coordinate-level <img src=image/13443798_05.gif>-adic expansions of the arithmetic may be obtained by employing <img src=image/13443798_05.gif>-adic expansion techniques to the arithmetic of points on the elliptic curve <img src=image/13443798_07.gif> in projective coordinates for <img src=image/13443798_08.gif> = 2, 3, ... In this paper, coordinate-level <img src=image/13443798_05.gif>-adic expansions of the arithmetic of points on <img src=image/13443798_09.gif> in projective coordinates are formulated, an algorithm for the computations is given illustrating the step-by-step process for computing point addition and doubling in <img src=image/13443798_09.gif> and its efficiency over the arithmetic of points on <img src=image/13443798_09.gif> in affine coordinates is described. This provides a systematic framework for performing elliptic curve arithmetic efficiently.

An asymptotic formula is derived for the summatory function <img src=image/13443653_02.gif>, where <img src=image/13443653_03.gif> denotes the total number of prime factors of <img src=image/13443653_16.gif> counted with multiplicity, and <img src=image/13443653_04.gif> is a multiplicative arithmetical function satisfying <img src=image/13443653_05.gif> for primes <img src=image/13443653_06.gif> and non-negative integers <img src=image/13443653_07.gif>, where <img src=image/13443653_08.gif> and <img src=image/13443653_01.gif> for <img src=image/13443653_09.gif>. The study builds on a rich history in analytic number theory, including classical results by Dirichlet on the divisor function <img src=image/13443653_10.gif>, and refinements using zeta-function estimates, as well as probabilistic approaches like the Erdős–Kac theorem extended to <img src=image/13443653_03.gif> (distinct primes) over <img src=image/13443653_11.gif>-free and <img src=image/13443653_11.gif>-full numbers. However, prior research has largely overlooked the multiplicity in <img src=image/13443653_03.gif> and its twisting by broad classes of multiplicative functions beyond divisors, particularly for square-full integers. The analysis covers three distinct cases: when <img src=image/13443653_04.gif> belongs to the subclass <img src=image/13443653_12.gif> (where <img src=image/13443653_13.gif> for all primes <img src=image/13443653_06.gif>); when <img src=image/13443653_04.gif> is in the broader class <img src=image/13443653_15.gif> but not in <img src=image/13443653_12.gif>; and when <img src=image/13443653_16.gif> is square-full with <img src=image/13443653_14.gif>. Examples of such functions include the number of non-isomorphic Abelian groups of order <img src=image/13443653_16.gif>, the number of square-full divisors of <img src=image/13443653_16.gif>, the divisor function <img src=image/13443653_17.gif>, and the <img src=image/13443653_07.gif>-fold divisor function <img src=image/13443653_18.gif>. The results are obtained using Dirichlet series <img src=image/13443653_19.gif>, which admit an Euler product decomposition due to multiplicativity, enabling analytic continuation via differentiation with respect to an auxiliary parameter <img src=image/13443653_20.gif>, contour integration, and estimates for the Riemann zeta function, as well as analytic continuation techniques. The following results were obtained: for case (i), <img src=image/13443653_21.gif>; for case (ii), <img src=image/13443653_22.gif>; and for square-full <img src=image/13443653_16.gif>, <img src=image/13443653_23.gif>. The work is theoretical in nature. The results of this study can be applied in further research in number theory, group theory, and discrete mathematics, with potential applications in algorithmic number theory (e.g., efficient computation of group orders) and cryptographic protocols relying on prime factorizations.

This paper introduces and explores a range of distance measures defined on multi-fuzzy sets, emphasizing both their mathematical foundations and applicability to real-world decision environments. Classical distance metrics such as Minkowski, Hamming, and Euclidean measures are extended to the multi-fuzzy context, and their behaviour is analysed at both the set and element levels. The proposed formulations are rigorously analyzed at both the set level and element level to capture variations in structure and similarity more precisely. The study further examines how these measures are affected when multi-fuzzy sets are transformed via crisp functions or adjusted using fuzzy weight matrices. The Minkowski distance in the original multi-fuzzy sets dominates or bounds the corresponding distance in the multi-fuzzy weighted sets via fuzzy matrix transformation. In addition to these classical extensions, this paper introduces new deviation-based and normalised measures aimed at quantifying unanimity and consensus within group decision-making processes. By extending classical statistical notions such as mean, variance, and standard deviation into the multi-fuzzy domain, the authors develop refined methods for assessing agreement among individual judgments. These are further strengthened through the use of weighted criteria to reflect varying importance. A numerical case study is provided to demonstrate the practical effectiveness of the proposed approach in real-world consensus evaluation. By improving the accuracy of collective decision-making models, the research contributes to transparent, equitable, and evidence-based decision support systems in fields such as education, healthcare, and policy analysis. The study is primarily theoretical and validated through a limited case study; future work may involve empirical validation across larger datasets or multi–fuzzy–neutrosophic extensions.

Fixed point theory constitutes a fundamental pillar of nonlinear analysis and has found extensive applications in mathematical modeling, optimization, computer science, and engineering. Classical results such as Banach's contraction principle have been generalized to various settings, including fuzzy metric spaces, cone metric spaces, and modular metric spaces. However, these frameworks often prove inadequate for modeling uncertainty involving indeterminacy and inconsistency. To address this limitation, neutrosophic fuzzy metric spaces (NFMSs) provide a powerful mathematical structure by integrating fuzzy distance measures with neutrosophic logic. In this paper, we establish several new fixed point theorems for single-valued mappings in complete neutrosophic fuzzy metric spaces under different generalized contractive conditions. The proposed contractions extend classical Banach-type, nonlinear, and rational-type contractions by incorporating neutrosophic fuzzy control functions. Using iterative techniques and properties of t-norms, we prove the existence and uniqueness of fixed points and demonstrate the convergence of the associated Picard iterative sequences. The obtained results significantly generalize and unify several existing fixed point theorems in fuzzy metric spaces, cone metric spaces, and modular metric spaces. An illustrative example is provided to validate the applicability of the main results. The primary contribution of this work lies in enriching the theoretical foundation of neutrosophic fuzzy analysis and offering a unified approach to handling uncertainty, vagueness, and inconsistency within fixed point theory. Although this study is mainly theoretical, the results have potential implications for optimization theory, decision-making models, and computational intelligence systems operating under indeterminate or conflicting information. Future research may focus on extending these results to multivalued mappings and their applications in real-world neutrosophic models.

This study presents the development and application of two semi-analytical methods—namely the Adomian Laplace Theorem and the Adomian–Kamal Theorem—for solving the Fractional Abel Differential Equation (FADE). Both approaches integrate the Adomian Decomposition Method (ADM) with distinct integral transforms to enhance accuracy and computational efficiency. The Adomian–Laplace method combines ADM with the Laplace Transform (LT), while the Adomian–Kamal method incorporates the Kamal Integral Transform (KIT), enabling improved handling of the non-local and long-memory characteristics inherent in fractional-order systems. Additionally, a fractional extension of the classical Euler method is implemented for comparative purposes. The methods are evaluated through two case studies, where approximate solutions are compared to exact solutions for various fractional orders α. Graphical analyses demonstrate that both semi-analytical methods yield results that perfectly overlap with exact solutions, indicating high accuracy and convergence. In contrast, the fractional Euler method exhibits reduced accuracy at lower fractional orders due to its limited capability of capturing memory effects. The findings highlight the superior performance and reliability of the Adomian–Kamal and Adomian–Laplace approaches for solving nonlinear FADEs, offering a robust framework for analytical and semi-analytical modeling in physics, engineering, and applied sciences.

Ignoring survey design features such as clustering, stratification, and unequal weighting can lead to underestimated standard errors (SEs) and misleading inference in regression models. This study compares three designconsistent variance estimators, Taylor linearization (TL), Fay's balanced repeated replication (BRR), and the Rao–Wu–Yue bootstrap, using Monte Carlo simulations based on the two-stage stratified structure of UNICEF's Multiple Indicator Cluster Surveys (MICS). Nine scenarios combine intraclass correlation (ICC = 0.01–0.10) with weight variability (<img src=image/13444009_01.gif>=0.2–1.0) to assess 95% coverage, SE calibration, and confidence interval (CI) width. Coverage was generally near nominal when clustering was weak to moderate (ICC ≤ 0.05), with mild under-coverage (about 90%) at ICC = 0.10 across methods. SEs were well calibrated (SE-ratios ≈ 0.93–1.03). CI width was driven primarily by weight heterogeneity, increasing markedly with larger <img src=image/13444009_01.gif>, whereas ICC had a smaller impact. In an application to 2018–2019 MICS data on childhood diarrhea, point estimates (odds ratios) were identical across methods; BRR and RWY bootstrap yielded slightly wider, more conservative CIs. Overall, TL is most efficient under moderate design effects, while replication methods offer greater robustness when clustering and weight dispersion are high, providing practical guidance for MICS-type analyses.

ANOVA-Simultaneous Component Analysis (ASCA) is a powerful approach for analyzing multi-factorial experimental designs with multivariate responses. However, standard ASCA methods assume homogeneous variance across experimental groups, an assumption often violated in realworld data, particularly in biological and chemical studies. This paper introduces Variance Normalized ASCA (VNASCA), a novel extension that addresses heterogeneous variance through a weighted least squares framework. We provide a comprehensive mathematical derivation of the VNASCA methodology, including its weight calculation strategies, permutation-based significance testing, and variance partitioning approach. The method incorporates robust options for weight estimation, regularization for numerical stability, and automatic effect selection for enhanced interpretability. Practical implementation guidelines for parameter tuning are provided, and the method's performance is validated through both controlled simulations and application to real agricultural data from maize trials in Ghana. The experimental results demonstrate that VN-ASCA provides improved prediction accuracy in the presence of heterogeneous variance while maintaining interpretability. The application to multi-environment maize trials in Ghana (50 plots, 100 traits, 4 locations) demonstrates practical utility, with adaptive VN-ASCA showing measurable improvements in root mean squared error and substantially stronger statistical evidence for environmental effects. These findings suggest that accounting for variance heterogeneity can enhance the detection and interpretation of factorial effects in complex multivariate datasets.

The study of Hopficity in Abelian groups has been largely motivated by the fundamental results of Baumslag, who proved that torsion groups are always Hopfian regardless of their cardinality, but left several questions open concerning torsion-free groups. Later, Corner addressed some of these questions by providing counterexamples showing that a direct sum of two Hopfian groups can be non-Hopfian, and that a group with an automorphism group of order two does not guarantee Hopficity. These results highlighted the need for new constructions to explore Hopficity in torsion-free Abelian groups. Our work introduces a new approach based on divisibility techniques, as it contributes to the understanding of free-torsion groups with respect to the Hopficity property, providing new insights into their structural properties and implications within group theory. Our analysis also demonstrates how divisibility properties, as well as the introduction of totally invariant subgroups and homomorphisms, can be used to establish Hopficity in specific Abelian groups, particularly those that are free-torsion. In order to reach all of this, we start by taking a group defined as an infinite direct sum of cyclic groups; then we construct a specific subgroup generated by two particular families of elements; and finally we show that this group is Hopfian through results from the theory of divisible subgroups.

This paper proposes an extensive finite element analysis of the one-dimensional Gray-Scott reaction-diffusion model (GSRDM), which is a fundamental framework for examining pattern formation in chemical and biological phenomena. A fully discrete numerical strategy was constructed utilizing the Galerkin finite element method (GFEM) for spatial discretization and the backward Euler (BE) scheme for temporal discretization. The nonlinear term is meticulously treated using a fully discrete formulation, preserving its authentic characteristics. The stability and convergence analysis of the discrete formulation is rigorously investigated by employing the error splitting and elliptic projection techniques with a special treatment for the nonlinear reaction terms. Numerical investigations employing a MATLAB script validated the predicted convergence rates and affirmed the precision of the proposed techniques. The impact of space and time-step refinements is examined comprehensively, supported by exact solutions and norm-based error analysis. A comparison with referenced works is discussed to demonstrate the effectiveness of the proposed scheme. This research offers a robust and adaptable framework for further studies on nonlinear reaction-diffusion systems.