Within the broader framework of quadri-partition neutrosophic soft bi-topological spaces (QPNSBTS), the concept of quadri-partition neutrosophic soft locally compact space (QPNSLCS) is introduced in this research. It strengthens the theoretical foundation for handling uncertainty in complex topological structures by demonstrating that local compactness, particularly when combined with the Hausdorff requirement, entails the existence of compact neighbors and compactness in subspaces. The key concepts and theorems illustrate how compactness can be effectively used in the context of neutrosophic soft sets, which are a more powerful way to handle unclear and ambiguous data in advanced mathematical and practical applications. Furthermore, a number of machine learning algorithms are used to explore the concept of a tangent similarity between two quadri-partition neutrosophic soft sets. Additionally, the current study includes a number of studies and visualizations to evaluate the effectiveness of different clustering algorithms and dimensionality reduction techniques. Each of the graphics in the findings illustrates a distinct method for viewing and comprehending complex data. The K-means++ initialization (Fig. 6.1) serves as an illustration of how the algorithm's initialization step improves clustering accuracy by choosing centroid (data points) that are widely distributed, reducing the likelihood of subpar clustering performance. More training is required since hidden units are only activated with low activations, according to restricted Boltzmann Machine (RBM) activation patterns (Fig. 6.2). Additionally, the Linear Discriminant Analysis (LDA) plots (Fig. 6.4) and Heatmaps (Fig. 6.3) might provide helpful details regarding the organization and segregation of the datasets. The discussion of the results, which can be devoted to their applicability in terms of clustering, dimensionality reduction, and feature learning, is based on these methods and the associated visual models.

The present work aims to extend the one-dimensional Hadamard fractional operator to a general multidimensional setting and study how this newly built operator acts between two Campanato spaces, Kp,µ (D) and Kq,ν (D). Alongside the boundedness statement itself, we pin down an explicit rate at which the operator norm grows with the diameter of D. Two features distinguish this study from prior boundedness results for fractional-type operators. First, the Campanato scale is strictly finer than the Morrey scale on which such questions are usually posed. Second, the Hadamard operator sits inside the Katugampola family as the degenerate case ρ → 0+, carrying a logarithmic kernel (log v σ)s−1 in place of the power kernels that dominate the classical theory – and it is exactly this logarithmic structure that drives the sharper estimates obtained below.

In this work, we establish several new variants of the Hermite–Hadamard integral inequality, together with various product-type extensions, by employing the Prabhakar fractional integral operator with three-parameter Mittag-Leffler kernels for mappings defined on harmonic sets. This unified operator generalizes several classical fractional operators, including Riemann–Liouville, Erdélyi–Kober, and Weyl, which are recovered as special cases. The results obtained herein constitute natural generalizations and significant improvements of existing results developed using classical integral operators. To demonstrate the validity and effectiveness of the obtained results, we present nontrivial numerical examples supported by graphical illustrations and tabulated comparisons for different choices of the fractional parameters. Furthermore, as an application of our main results, we derive explicit error estimates for the trapezoidal rule on harmonic intervals.