DERIVATIVES THROUGH PROBES IN REGULAR GEOMETRIC OBJECTS: A DIMENSIONAL EXPLORATION FOR q-SETS IN TIME SCALE CALCULUS
Fractals, cilt.34, sa.5, 2026 (SCI-Expanded, Scopus)
- Yayın Türü: Makale / Tam Makale
- Cilt numarası: 34 Sayı: 5
- Basım Tarihi: 2026
- Doi Numarası: 10.1142/s0218348x26500271
- Dergi Adı: Fractals
- Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Compendex, INSPEC, zbMATH
- Anahtar Kelimeler: Algebra, Deformation, Fractional Analysis, Inference, Time Scale Calculus
- Uşak Üniversitesi Adresli: Evet
Özet
The tools of stochastic geometry facilitate the extraction of sample forms from regular geometric objects. In this context, probes serve as instruments for establishing sampling forms characterized by sub-dimensions, such as points, lines, and areas associated with these objects. This study employs known formulas pertaining to these geometric entities to propose a differentiation framework rooted in the reconstruction of regular geometric objects using the corresponding probes. The definition of time scale calculus, recognized as a generalization of the linear derivative, is applied to the differences observed between the geometric objects and the probes. Following the establishment of these differences, we derive nonlinear derivatives based on a specific form of Jackson’s q-derivative, which serves as a specialized instance of time scale calculus. Within the linear framework of derivatives, the convex combination of two points is utilized to generate a new point through the proposed derivative within the linear set. This linear perspective implies that the derivative encapsulates a point that can similarly be generated by the derivative in the nonlinear set. Consequently, we arrive at compelling justifications for the definition of derivatives within a nonlinear context, thereby offering alternative formulations. Furthermore, a generalized cosine theorem is derived to delineate various nonlinear forms of derivatives, allowing for the proposition of correlations based on this theorem. Once the derivatives are established, corresponding integrals are introduced, as integrals are inherently the inverses of derivatives, and vice versa. The study also explores applications related to deformed numbers, the generalization of maximum likelihood estimation derived from the proposed entropy functions, Fisher information, and the delta method based on Taylor expansion, all grounded in the proposed derivatives. These applications are intended to enable alternative precise measurements and modeling of phenomena when reality is represented in this manner.