One third of people with epilepsy (PwE) live with drug-resistant epilepsy (DRE), a condition in which seizures persist despite adequate medication. Epilepsy is now understood as a network disorder that disrupts large-scale neuronal coordination, and alterations in the scale-free properties and fractal organization of brain dynamics may emerge as signatures of pathological brain functions. This study investigates whether the temporal sequence of EEG microstates (the brief, quasi-stable topographies that reflect ongoing functional configurations) shows altered fractal properties in DRE compared with drug-responsive epilepsy (nDRE) and whether these properties change after therapeutic neuromodulation. We tested the following hypotheses: 1) fractal features of brain dynamics are different between people with DRE and people with drug-responsive epilepsy (nDRE), thus qualifying them as a potential diagnostic biomarker; 2) fractal features of brain dynamics can be modulated in people with DRE, thus qualifying them as a potential response biomarker.

Resting-state EEG (eyes closed) was examined in 60 DRE and 60 nDRE patients and in a subgroup of 10 DRE patients recorded before and after Vagus Nerve Stimulation (VNS). The study was approved by the Ethics Committee ‘Lazio Area 2’ (number 99.24CET2 CBM, April 11, 2024). Microstate sequences were extracted, and their temporal structure was characterized using scale-free metrics, including the Higuchi Fractal Dimension and Hurst exponent, which quantify short-time complexity properties and long-range dependencies. While conventional microstate metrics and transition patterns did not differentiate DRE from nDRE, fractal analyses revealed significantly higher scale-free disruption in DRE, indicating less self-similar and more erratic state trajectories. Notably, similar fractal shifts were observed after VNS and accompanied clinical improvement, suggesting that fractal markers are sensitive to therapeutic modulation. These findings indicate that the scale-free structure of brain-state sequences may capture fundamental alterations in cortical dynamics associated with DRE. Identifying fractal biomarkers for DRE could support earlier diagnosis, reduce disease burden, and advance precision approaches tailored to individual network dysfunction.

This study presents a novel application of fractal geometry and complex systems theory to a mufti-layered socioeconomic dataset: NASA's GRDI v1, a global high-resolution (≈1 km) index of multidimensional deprivation. We present evidence that the spatial architecture of multidimensional deprivation is self-similar across scales, exhibiting a robust fractal geometry. Analyzing data from the Global Relative Deprivation Index (GRDI v1), we compute the box-counting dimension across six continental regimes. The analysis reveals a mean fractal dimension of D = 1.60 ± 0.40, demonstrating significant regional heterogeneity (ANOVA F = 122.27, p = 7.28 × 10⁻¹⁰), from densely clustered deprivation in South Asia (D = 1.904) to sparse, fragmented patterns in South America (D = 0.781). This power-law scaling is exceptionally strong (mean R² = 0.997) and is validated within 5% tolerance by correlation dimension analysis in 90% of regions.

The observed scaling is not an artifact of methodology. Fractal dimensions are stable across deprivation thresholds (coefficient of variation = 3.40%) and are statistically preferred over log-normal or stretched-exponential alternatives (likelihood ratio test, p < 0.001). Controlling for spatial autocorrelation, we identify a significant association between fractal dimension and governance indicators (Mantel test r = 0.73, p = 0.003).

Theoretically, these findings characterize deprivation as a spatially embedded complex system with a non-trivial topological dimension (1 < D < 2). This indicates a sparse, hierarchical network structure that is nested across scales, implying that local interventions may propagate through self-similar pathways.

Practically, D serves as a scale-invariant metric for comparative analysis. It delineates distinct policy arenas: regions with D > 1.7 necessitate multi-scale strategies addressing hierarchical clustering, while regions with D < 1.2 may be more responsive to targeted, nodal interventions. We further demonstrate that higher D correlates with slower recovery from economic shocks (r = -0.61, p= 0.02), suggesting that fractal complexity quantifies systemic fragility.

We argue that fractal mathematics provides an essential analytical tool for dismantling persistent deprivation structures, offering the first rigorous proof that poverty functions as a complex adaptive system with scale-invariant properties.

Surface topography characterization increasingly relies on areal parameters (ISO 25178) to relate manufacturing processes to functional performance (e.g., wettability, adhesion, tribology). Yet, most roughness parameters remain scale-dependent, making multiscale frameworks essential for describing complex, fractal-like surfaces. In this work, we introduce a standard-friendly multiscale approach based on the interfacial area ratio Sdr (ISO 25178-2) iterated across scales using low-pass Gaussian filtering (ISO 16610), yielding an area–scale curve that quantifies surface complexity through relative area development. This protocol provides a practical alternative to the well-established Richardson Patchwork (triangular tiling) area-scale method used in standards, while relying exclusively on widely implemented standardized operators (Gaussian filter + ISO parameter), enabling straightforward deployment in conventional surface metrology software.

We validate the approach on grit-blasted TA6V surfaces produced under controlled variations of blasting pressure and media, showing that the multiscale Sdr procedure reproduces Patchwork trends over the investigated scale ranges and remains discriminant with respect to process conditions.

To objectively identify the most informative observation scale, we further propose an uncertainty-based scale selection strategy using bootstrap regression, highlighting a relevant cut-off length around 120 µm for capturing the pressure–topography relationship across media.

Overall, this work bridges fractal-inspired area-scale analysis with ISO-compliant metrology practice, providing a robust and accessible route to multiscale characterization and to process–topography interaction studies.

The heterogeneity of pore systems across micro- to macro-scales critically controls gas storage and transport in shale, yet it remains challenging to quantify it accurately, especially in complex marine–continental transitional facies. Here, we introduce a novel pore-volume-weighted comprehensive fractal dimension (Dₜ) model that integrates high-pressure mercury intrusion, low-temperature nitrogen adsorption, and carbon dioxide adsorption data. This approach overcomes the limitations of traditional single-fractal models by enabling a unified quantification of pore complexity from micro- to macro-pores. Applied to the Upper Permian Longtan Formation shale of the South Yellow Sea Basin, our model indicates that mesopores dominate the pore network. Samples with high total organic carbon (TOC) content exhibit distinctive dual-segment fractal features in nitrogen adsorption data, revealing the controlling role of organic matter on pore complexity. Furthermore, the Dₜ shows strong positive correlations with TOC, quartz content, and the Brunauer–Emmett–Teller specific surface area, suggesting that organic richness and brittle minerals synergistically govern pore system complexity. This study demonstrates that the Dₜ model provides a robust mathematical framework for full-scale pore characterization. Our findings offer a fractal-based theoretical foundation for predicting reservoir quality in transitional shale systems. Our understanding of the influence of geological conditions on pore systems has been deepened by applying fractal theory.

Fractal image ompression (FIC) provides the distinctive advantage of resolution independence, enabling deep zooming and super-resolution without the pixelation artifacts commonly observed in traditional Discrete Cosine Transform (DCT)-based methods. Despite these benefits, the high computational cost of exhaustive domain block searching has limited the practical adoption of FIC for real-time video applications. In this paper, we propose a novel high-performance architecture for fractal video compression. The proposed approach incorporates a variance-based intelligent search heuristic to substantially reduce the domain search space, along with a massively parallel GPU kernel for efficient affine block matching. According to experimental findings, the suggested approach maintains a high structural similarity index measure (SSIM) and supports resolution-independent zooming capabilities while achieving notable performance improvements over CPU-based implementations.

Despite its theoretical benefits, the computational difficulty of FIC has severely limited its practical usage, especially for video compression. To find the optimal affine match for each range block, the encoding method necessitates a thorough search across a vast pool of domain blocks. The computational cost of this brute-force matching is on the order of O(N_r x N_d), where N_R and N_D represent the number of range and domain blocks, respectively. In conventional CPU-based systems, this complexity makes real-time encoding unfeasible. To overcome this fundamental bottleneck, this paper proposes a high-performance fractal video compression framework that leverages both algorithmic optimization and hardware acceleration.

By combining intelligent search heuristics with GPU parallelism, the proposed approach substantially accelerates fractal encoding while maintaining high reconstruction quality and preserving the intrinsic resolution-independent benefits of fractal compression.

The study demonstrated the successful implementation and testing of a CUDA-Accelerated Intelligent Fractal Video Compressor that resolves the traditional performance limitations of fractal image compression (FIC). Our key achievement came from combining an intelligent variance-based heuristic with a parallel GPU kernel that we optimized highly.

Meditation has been proposed to alter large-scale brain dynamics, but its impact on neural complexity remains subtly yet incompletely understood. In this study, we investigated how three contemplative traditions, Himalayan Yoga Tradition, Vipassana, and Isha Shoonya Yoga, modulate electroencephalography (EEG) complexity, quantified via Higuchi’s Fractal Dimension (HFD). Based on the Entropic Brain Hypothesis, we hypothesized that regular meditative practice would be associated with increased EEG complexity relative to a non-meditative condition. We used a publicly available EEG dataset. In the original study, written informed consent was obtained, and the protocol was approved by the local MRI (India) ethics committee and the UC San Diego IRB (#090731). EEG data were recorded from 48 participants (16 per meditation group) and a non-meditating control group during two 20-minute blocks: meditation (including an initial breath-focused phase) and instructed mind-wandering. After standard preprocessing (filtering, artifact rejection, ICA) and segmentation into short overlapping epochs, HFD was computed for each electrode. The parameter Kmax was optimized by inspecting the HFD–Kmax curve across subjects, yielding a stable plateau around Kmax = 10, which was adopted for all subsequent analyses. Cluster-based paired permutation tests were used to compare breath focus, full meditation, and mind-wandering within each group. Across meditation traditions, we observed a robust increase in HFD during meditation relative to instructed mind-wandering, with the strongest effects during the breath-focus phase, indicating higher brain signal complexity in meditative states. These findings support the Entropic Brain Hypothesis by suggesting that meditation shifts neural activity toward regimes of enhanced complexity, and highlight EEG fractal metrics as sensitive markers of meditation-induced changes in brain dynamics.

The hybrid control law incorporating adaptive backstepping is suggested for the synchronization of two identical modified piecewise Rossler models. The suggested approach gives rapid, high rigorous chaos synchronization in the presence of external disturbances and unknown system parameters. The adaptive backstepping control approach is established to stabilize the dynamics of synchronization error occurring from the conflict (mismatch) between the response and drive systems. The unknown external disturbance and unknown system parameters are determined by the adaptive backstepping law in such a way that the whole system, with the controller, stabilizes. Performance of the suggested law is examined for a numerical solution and compared against frequently utilized chaos synchronization approaches including active control, optimal control, and adaptive backstepping control approach. It was investigated
for the aspect of chaos synchronization and synchronization errors. The synchronization of models via hybrid adaptive backstepping for the comparison of different derivatives with a novel MABC fractional operator is exhibited. The MABC fractional operator is a nonsingular operator obtained by integration by parts of the standard fractional operator with the Mittag–Leffler kernel. An effective approach is suggested for stabilizing chaos. Computer simulation results demonstrate the superior performance of the hybrid control approach over the state of the art for the synchronization of modified Rossler models.

Helianthus annuus exhibits complex morphological structures, including leaves, stems, and flower heads, which often display self-similarity and multiscale organization. Understanding these patterns is essential for insights into plant development, growth dynamics, and ecological adaptation. Fractal geometry and fractional calculus provide robust frameworks for analyzing such structural complexity and modeling the spatial and temporal dynamics of plant morphology. This work explores the application of fractal measures and fractional-order modeling techniques to Helianthus annuus morphology, emphasizing their ability to capture multiscale patterns, symmetry, and memory effects in growth processes. Different approaches reported in the literature, including fractal dimension estimation and fractional modeling, are conceptually compared in terms of biological interpretability, analytical robustness, and potential applications in plant growth studies. Special attention is given to how non-integer order operators can represent adaptive structural development, variability in leaf and flower arrangement, and complex spatial correlations. Furthermore, the discussion includes potential implications for agricultural optimization and ecological research. The synthesis identifies key methodological trends, advantages, and current limitations of fractal and fractional approaches in plant morphogenesis research. By integrating theoretical frameworks with observed sunflower structures, this work provides a coherent perspective on plant morphology and supports the broader use of advanced mathematical tools in quantitative botany.

Various approches for constructing fractional cosmological models with additional dynamical degrees of freedom are presented. Focusing on one of these approaches, the corresponding fractional equations of motion are explicitly derived. In this fractional framework, the standard continuity equation of the corresponding relativistic cosmological model is no longer preserved, as energy is continuously exchanged between the fractional sector and the additional dynamical components. To address this issue, effective energy density and pressure are consistently defined in such a way that an effective continuity equation is satisfied. This procedure plays a crucial role not only in obtaining an appropriate dynamical system for the background evolution, but also in the systematic derivation of the perturbation equations.

The differences, advantages, and limitations of our fractional framework are examined in direct comparison with the corresponding standard cosmological models, both at the level of the field equations and their exact solutions. This comparative analysis is carried out not only at the background level, but also within first-order cosmological perturbation theory. In particular, the fractional generalization of the Mukhanov–Sasaki equation is derived, providing a consistent description of scalar perturbations in the presence of fractional effects and memory contributions. The resulting framework offers a coherent extension of relativistic cosmology in which fractional dynamics naturally influence both the background evolution and the perturbative sector.

This study investigates a generalised nonlinear fractional diffusion–advection equation incorporating concentration-dependent diffusion and nonlinear advection terms. The model $\mathcal{D}_t^\beta u(x,t) = \frac{\partial}{\partial x} \left( f(u) \frac{\partial u}{\partial x} \right) - \frac{dK}{du} \frac{\partial u}{\partial x}$, where, \( \mathcal{D}_t^\beta \) represents the Caputo derivative of the fractional order \( \beta \in (0,1] \), and \(f(u), \frac{dK}{du} \) are nonlinear functions. This formulation incorporates a concentration-dependent diffusion coefficient, allowing diffusive behaviour to vary with the state variable, and a nonlinear advection term that more realistically accounts for convective effects. The mathematical model is expressed using the Caputo fractional derivative, which is particularly suitable for problems with physically meaningful initial conditions and memory effects. To establish mathematical well-posedness, sufficient conditions for the existence and uniqueness of solutions are derived by applying the Banach fixed-point theorem. This analysis guarantees that the problem admits a unique solution within an appropriate functional framework, providing a solid theoretical foundation for further investigation. For constructing approximate analytical solutions, the homotopy perturbation method (HPM) is employed. This technique yields solutions in the form of rapidly convergent series without requiring small parameters or linearizing the governing equation. Several illustrative examples demonstrate the effectiveness, accuracy, and simplicity of the proposed approach. The obtained results confirm the applicability of HPM to nonlinear fractional diffusion–advection problems and generalise earlier studies on fractional diffusion equations and Burgers-type equations, offering a unified framework for analysing a wide range of nonlinear fractional models.