Fractal and Fractional

We investigate the existence and multiplicity of positive solutions for an $\omega$-Riemann–Liouville fractional differential equation on an infinite interval supplemented with nonlocal boundary conditions containing Riemann–Stieltjes integrals and $\omega$-Riemann–Liouville fractional derivatives of various orders. In the proof of our main results we use the Guo–Krasnosel’skii fixed point theorem and the Leggett–Williams fixed point theorem.

The performance of an optimization algorithm can be limited when the same search behavior is maintained throughout the optimization process. Fractional Newton-Derived Optimization (FNDO) uses a fixed fractional order and does not adapt its main search mechanisms during the run. To address this limitation, this study introduces SA-FNDO, an adaptive extension of FNDO that retains the original fractional order while incorporating archive-assisted guidance based on JADE, covariance-based crossover derived from CoBiDE, and CMA-ES refinement around the current best solution. SA-FNDO is evaluated on the CEC2017 and CEC2022 benchmark suites and on three static 3D UAV path-planning environments. The ablation study shows that archive guidance and CMA-ES refinement provide the clearest contributions, while the effect of covariance crossover is more problem-dependent. Across the CEC test conditions, SA-FNDO obtains the best overall mean rank among the nine compared methods, ranking first on CEC2017 and CEC2022 at $D=10$ and second on CEC2022 at $D=20$. In the UAV case study, SA-FNDO obtains feasible paths in all 90 runs and achieves the lowest feasible-run median geometric cost in all three environments. These results support SA-FNDO as an effective extension of the publicly available FNDO implementation under the tested conditions.

Fractal analysis has become a frequently used technique in sensory physiology for the evaluation of signal complexity, with potential applications ranging from the analysis of retinal vascular architecture to the quantification of changes in electroencephalographic recordings. Features such as fractal dimension and lacunarity may be able to detect discrete changes in both one-dimensional and two-dimensional signals associated with numerous physiological processes and disease development. However, their interpretation and translational potential are constrained by various limitations. In this concise review, we critically explore current applications of fractal analysis across major sensory domains and focus on the challenges that limit reproducibility and biological interpretation. We particularly focus on the biological ambiguity of fractal dimension and related measures, the non-equivalence of commonly used fractal metrics, acquisition-related variability, small datasets, and insufficient technical validation. Finally, we discuss issues related to the integration of fractal metrics into contemporary machine learning models, including challenges related to generalizability, overfitting, and poor explainability.