A Novel Approach to Fractal Generation through Strong Coupled Fixed Points in Intuitionistic Fuzzy Metric Spaces
Bibliographic record
Abstract
In this manuscript, we explore the concept of strong-coupled fixed points in the context of intuitionistic fuzzy metric spaces (IFMS). Our approach is grounded in the idea of intuitionistic fuzzy contractive couplings (IFCCs), which provide a framework for understanding fixed points in fuzzy settings. We begin by introducing a novel formulation of coupling, which combines the principles of coupled fuzzy contractions with cyclic mappings. This combination leads to a more generalized and effective method of identifying strong-coupled fixed points, extending previous results in fuzzy metric spaces. A key contribution to this paper is the proof of the existence of a unique strong-coupled fixed point. We establish this result through rigorous theoretical analysis and provide a corollary that strengthens the foundation of our work. Several non-trivial examples are presented to demonstrate the applicability of the theory and the robustness of the strong-coupled fixed point in various scenarios. Additionally, we present a practical application of our findings: the construction of a strong-coupled fractal set within the framework of intuitionistic fuzzy metric spaces. This is achieved by applying an intuitionistic iterated function system (IIFS), which is based on a family of intuitionistic fuzzy contractive couplings. The fractal generation process is illustrated through several examples, demonstrating the theoretical results in action. To further solidify the applicability of our approach, we introduce an intuitionistic fuzzy version of the Hausdorff distance between compact sets, a crucial tool in measuring the "closeness" of sets within the intuitionistic fuzzy context. Several examples are provided to clarify the fractal generation process, showing how the intuitionistic fuzzy metrics and couplings contribute to the creation of self-similar fractals. This work not only enhances the understanding of fixed points in intuitionistic fuzzy spaces but also provides new insights into their application in fractal geometry, offering both theoretical advancements and practical tools for future research in this area.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".