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.

Effective data partitioning is known to be crucial in machine learning. Traditional cross-validation methods like K-fold Cross-Validation (KFCV) enhance model robustness but often compromise generalisation assessment due to high computational demands and extensive data shuffling. To address these issues, the integration of the Simple Random Sampling (SRS), which, despite providing representative samples, can result in non-representative sets with imbalanced data. The study introduces a hybrid model, Fusion Sampling Validation (FSV), combining SRS and KFCV to optimise data partitioning. FSV aims to minimise biases and merge the simplicity of SRS with the accuracy of KFCV. The study used three datasets of 10,000, 50,000, and 100,000 samples, generated with a normal distribution (mean 0, variance 1) and initialised with seed 42. KFCV was performed with five folds and ten repetitions, incorporating a scaling factor to ensure robust performance estimation and generalisation capability. FSV integrated a weighted factor to further enhance performance and generalisation. Evaluations focused on mean estimates (ME), variance estimates (VE), mean squared error (MSE), bias, the rate of convergence for mean estimates (ROC ME), and the rate of convergence for variance estimates (ROC VE). The results indicated that FSV consistently outperformed SRS and KFCV, with ME values of 0.000863, VE of 0.949644, MSE of 0.952127, bias of 0.016288, ROC ME of 0.005199, and ROC VE of 0.007137. These results demonstrate the superior accuracy (reduced error and bias) and reliability of FSV (stable performance with increasing sample size and repeated trials). Moreover, since FSV reduces the computational demands of repeated KFCV while preserving representativeness, it becomes particularly well suited to environments with constrained resources and large-scale datasets. Hence, FSV offers a practical solution for improving model evaluation where both efficiency and accuracy are critical.

This paper presents a detailed numerical analysis of a particular class of heat equations. To approximate the solutions, we develop a hybrid computational framework that combines the Haar wavelet collocation method with finite difference techniques. In the proposed approach, finite difference schemes are first applied to discretize the time variable, thereby reducing the original problem to a semi-discrete system. Subsequently, Haar wavelets defined on a uniform spatial grid are employed to approximate the spatial derivatives, allowing for efficient representation of both smooth and non-smooth solution profiles. Within the framework of Sobolev spaces, rigorous proofs of stability and convergence are provided, ensuring the reliability of the method from a theoretical standpoint. To validate the practical performance of the proposed scheme, extensive numerical experiments are conducted across a variety of test problems. The accuracy of the method is quantitatively assessed using both the <img src=image/13443061_01.gif> and <img src=image/13443061_02.gif> error norms. The results consistently demonstrate that the method delivers high accuracy while maintaining strong computational efficiency. Furthermore, the computed solutions exhibit excellent agreement with exact analytical solutions, thereby confirming the robustness and effectiveness of the hybrid scheme for solving heat equations. In addition, the findings of this research make several contributions to the field of numerical analysis of partial differential equations.

This study investigates the representation theory of the multiplicative group of a finite nearfield, with a specific focus on the group <img src=image/13440431_01.gif>, arising from Dickson nearfields. Nearfields, generalizations of fields lacking one of the two distributive laws, pose interesting challenges in algebraic structures due to their noncommutativity and asymmetry. Their multiplicative groups, particularly those derived from Dickson nearfields, have a metacyclic structure that allows for intricate analysis using representation theory. The purpose of this research is to construct and analyze the character tables of these multiplicative groups. We explore foundational concepts such as Schur's lemma, Maschke's theorem, and orthogonality relations, and apply them within the context of finite nearfields. A combination of theoretical and computational methodologies is employed. Specifically, we make use of the Magma algebra system to build and verify character tables for <img src=image/13440431_01.gif> for small values of <img src=image/13440431_02.gif> and <img src=image/13440431_03.gif>, including detailed derivations for <img src=image/13440431_04.gif> and <img src=image/13440431_05.gif>. The principal results include explicit character tables, classifications of conjugacy classes, degrees of irreducible representations, and a generalization framework for any pair (<img src=image/13440431_06.gif>). The research demonstrates that the multiplicative groups <img src=image/13440431_01.gif> are not only metacyclic but also nilpotent and solvable under specific conditions. It further establishes structural conditions that determine the conjugacy class partitioning and representation degrees. Our conclusions highlight the deep connection between nearfield algebraic properties and finite group representation theory. This work contributes to the theoretical development of group representations in non-classical algebraic settings and opens doors to applications in geometry, coding theory, and cryptography. Practically, the use of Magma facilitates automated representation analysis, and socially, this research promotes a broader understanding of non-traditional algebraic systems. While the current study is computationally verified for small instances, future work will aim to optimize algorithms for handling larger <img src=image/13440431_01.gif> structures, enhancing scalability and applicability.

The primary objective of this research is to introduce and analyse the concept of the static index number in the context of directed and total directed graphs. This study focuses on exploring the structural properties of two newly defined components, the engender edge set and the sturdy set, which play a crucial role in characterizing connectivity within directed graphs. A directed graph is said to be a minimally connected sub digraph, and if we remove any one edge from the graph, then the graph will be disconnected. The minimum cardinality of a set <img src=image/13443264_01.gif> is said to be the Engender edge set of <img src=image/13443264_05.gif> if (i) <img src=image/13443264_02.gif> (ii) <img src=image/13443264_03.gif> is a minimally connected subdigraph. The set of edges <img src=image/13443264_04.gif> is called a sturdy set of <img src=image/13443264_05.gif> and the minimum cardinality of a sturdy set is called a static index number of a directed graph <img src=image/13443264_05.gif> and it is denoted by <img src=image/13443264_06.gif>. This parameter serves as a novel measure of structural stability and minimal connectivity in directed graphs. In this work, we establish several theoretical results and derive bounds for the static index number of both directed and total directed graphs. Furthermore, we examine its relationship with other graph-theoretic parameters, particularly the out-degree and in-degree domination numbers. The findings provide deeper insight into the interplay between domination and minimal connectivity in directed networks and contribute to the broader understanding of graph invariants in digraph theory.

We propose a Bayesian framework for regression modeling that uses Pearson Type VII Processes (P7Ps) as an adaptable generalization of Gaussian Processes (GPs). The P7P is a scale mixture of Normal and Gamma distributions. It has flexible heavy-tailed properties that enhance its robustness against outliers and render it suitable for modelling heavytailed data. The P7P framework outperforms GP functions by providing explicit control over tail behavior and variance modulation via the shape and scale parameters. These factors dictate the weight of the tails and the overall dispersion of the process, facilitating a seamless transition between Gaussian and heavy-tailed regimes, thus improving modelling flexibility and robustness. We derive the predictive distribution for new data points and use the Laplace approximation for efficient posterior inference under the Pearson Type VII (P7) prior, which ensures scalability while maintaining analytical tractability. The Pearson Type VII Process for regression (P7PR) outperforms Gaussian Process for regression (GPR) in terms of resilience, adaptability and predictive accuracy, especially in challenging scenarios with heavy-tailed noise and outliers, as evidenced using comparative analysis based on Mean Square Error (MSE), Mean Absolute Error (MAE) and predictive log-likelihood (PLL), using both simulated and real-world datasets. These results validate the benefits of heavy-tailed priors in modelling non-Gaussian distributions and illustrate the resilience of P7PR as a Bayesian substitute for conventional GPR. Although the proposed P7PR framework exhibits robustness and strong predictive accuracy, its computational cost increases remarkably as the size of the datasets, the complexity of the model, and the posterior space expand. Consequently, in order to preserve scalability without compromising performance, practical implementations may necessitate moderate model sizes or approximate inference methods.

Conditional Autoregressive (CAR) models have been widely used in various disciplines, including epidemiological studies. The application of the CAR model in epidemiological studies is often associated with the relative risk of an infectious disease. This relative risk value can be estimated using the CAR models. Here, we evaluate four commonly used CAR models: the Intrinsic CAR, the Besag-York-Mollié CAR (BYM CAR), the BYM-modified CAR (BYM2 CAR), and the Leroux CAR (LCAR). To estimate CAR model parameters, Bayesian inference and the Integrated Nested Laplace Approximation (INLA) concept are used. The selected model was then used to model the number of dengue hemorrhagic fever (DHF) cases in Central Java Province in 2024. To support this analysis, we used 50 datasets simulated for each sample size (n), ranging from 10 to 100. The results of the study showed that of the four models compared, the best model was BYM2. This model was then used to model DHF cases in 2024 in Central Java Province. The research findings indicate the necessity of controlling population density, optimizing the role of medical personnel, and preparing for increased rainfall to curb the spread of dengue fever. Comprehensive detection and control measures through medical facilities are also required. Meanwhile, based on the coefficient of the altitude variable in the model, altitude has a positive influence on the number of dengue fever cases. Therefore, the conflicting conclusions between the model results and the medical perspective require data verification and further study of this variable.

This study presents a comprehensive analytical investigation of a four-server queueing system, modeled as a priority feedback queue network operating under stochastic conditions. The system architecture comprises a primary server that incorporates a priority mechanism and is connected to three parallel servers. Customers, divided into low- and high-priority groups, arrive at the primary server according to a Poisson process. Service times at all servers are assumed to follow exponential distributions. After completion of service at any server, a customer will either exit the system or re-enter for further service with a different feedback probability, which reflects a realistic service satisfaction mechanism. This analysis evaluates explicit expressions for time-independent state probabilities and derives steady-state performance metrics of the network, including server utilization, average queue lengths, waiting times, and throughput by application of generating function techniques and the calculus-based resolution of the associated differential-difference equations. Extensive numerical simulations are carried out to validate the theoretical results and to delve into the impact of different parameters on various performance indicators, which gives information to system administration in order to reduce congestion. This study yields actionable insights into the optimal allocation of resources, aiming to minimize delays and maximize throughput under varying stochastic conditions. Furthermore, the behavioral analysis of the model underscores the notable effects of priority and feedback mechanisms on the system achievement measures and provides a foundation for optimization strategies in multi-class service environments. The findings have significant implications for the design and optimization of complex queueing networks in telecommunications, manufacturing systems, and other service-oriented industries where feedback and priority-based processing are prevalent.

The Gaussian distribution and its related functions are widely used in statistics, engineering, and physics, including hypothesis testing, quality control, and digital communications. As the standard normal cumulative distribution function (CDF) lacks a closed-form, accurate approximations are essential for practical applications. This study presents new, highly accurate improvements to existing approximations of the standard normal distribution function while preserving simplicity. The proposed methodology refines current formulas by introducing additional parameters and systematically optimizing them using R-based iterative procedures guided by the Kolmogorov-Smirnov statistic to minimize maximum and mean absolute errors. Evaluations across a dense grid confirm the precision and robustness of the proposed models. Results demonstrate significant improvements, with one enhancement reducing the maximum error by a factor of 4.86 compared to previous models, while other formulas show notable accuracy gains. The improved models remain straightforward to implement, making them suitable for real-time engineering and applied statistical computations. A practical application is demonstrated by calculating the bit error rate (BER) in Binary Phase Shift Keying (BPSK) systems under Gaussian noise, where the proposed approximations closely replicate the exact BER curve, confirming their effectiveness in telecommunications system analysis. These improvements provide engineers and scientists with precise, efficient tools, supporting modeling, risk assessment, and system reliability in applications requiring accurate Gaussian evaluations.

This paper investigates the oscillatory behavior of solutions to a class of nonlinear differential equations on the Heisenberg group, where the dynamics are governed by the interplay between superlinear and sublinear terms. The Heisenberg group, as a fundamental example of a non-commutative Lie group with sub-Riemannian geometry, offers a natural and rich framework for analyzing such equations, which frequently arise in mathematical physics and geometric analysis. To establish oscillatory criteria, we employ a combination of the Riccati transformation and the integral averaging method. The Riccati technique enables the reformulation of the original equation into a form suitable for qualitative analysis, while the integral averaging approach helps derive sufficient conditions by capturing the average behavior of the nonlinear terms and coefficients over certain intervals. This dual approach allows us to rigorously examine how the interaction between superlinear and sublinear growth influences the oscillatory nature of solutions. We derive new sufficient conditions that guarantee the oscillation of all solutions under appropriate structural assumptions. These results generalize and extend classical oscillation criteria to the subelliptic equation of the Heisenberg group. To demonstrate the applicability and sharpness of our results, several illustrative examples are presented. The findings contribute to the broader understanding of oscillation theory in non-Euclidean space and pave the way for future research in nonlinear analysis on Lie groups.