Alluaudite-type materials are systematically attracting more attention as prospective cathode materials for sodium ion batteries. In this paper, we strove to optimize various synthesis parameters of three alluaudite compositions (Na${}_2$Fe${}_3$(PO${}_4$)${}_3$—FFF, Na${}_2$VFe${}_2$(PO${}_4$)${}_3$—VFF, and Na${}_2$VFeMn(PO${}_4$)${}_3$—VFM) to obtain nanostructured alluaudite-type glass-ceramics with high phase purity. We systematically investigated the role of the chemical reactions, temperature and time of melting, cooling rate, and reducing factors on the quality of the final products. A detailed synthesis protocol along with X-ray diffractometry, thermal analysis, scanning electron microscopy imaging, and electrical conductivity measurements (with impedance spectroscopy) are reported. As a result, a significant increase of the conductivity was observed in the nanomaterials. The highest value was reached for the VFF composition and was equal to $6\times {10}^{-4}$ S/cm at room temperature.

The tribological characteristics of a below 1 mm micro ball bearing comprising steel disc and cages coated with thin copper and silver films were investigated. Electroplating and laser cutting were used to manufacture used elements. Friction was measured using a linear stage and an adapted version of a friction-loop method. The obtained results show an interesting relationship between the geometric properties of the micro scale thrust bearing and their performance and operational stability, which can be correlated to similar relationships observed in the macro scale. The most optimal design of the bearing showed stable operation, with the simplified rolling resistance coefficient in the range 0.002 to 0.003, independently of applied load, which was in range 150 mN to 1500 mN. The possibility of creating easily manufacturable micro ball bearings with a low rolling resistance coefficient comprised solely of cheap and sturdy elements was shown.

Alluaudite-type materials are systematically attracting more attention as prospective cathode materials for sodium-ion batteries. It has been demonstrated that optimized thermal nanocrystallization of glassy analogs of various cathode materials may lead to a significant increase in their electrical conductivity. In this paper, three alluaudite-like glasses (Na${}_2$Fe${}_3$(PO${}_4$)${}_3$—FFF, Na${}_2$VFe${}_2$(PO${}_4$)${}_3$—VFF, and Na${}_2$VFeMn(PO${}_4$)${}_3$—VFM) were synthesized and subjected to an optimized thermal nanocrystallization. This procedure resulted in nanostructured samples with increased electrical conductivity at room temperature: $5\times {10}^{-7}$ S/cm (FFF), $7\times {10}^{-5}$ S/cm (VFM), and $6\times {10}^{-4}$ S/cm (VFF). The nanocrystalline microstructure was also evidenced by ultra-high-frequency impedance spectroscopy (up to 10 GHz) and proposed electrical equivalent circuits. Prototype electrochemical cells were assembled and characterized with voltage cutoffs of 1.5 and 4.5 V. The electrochemical performance was, however, modest. The gravimetric capacity varied between the studied materials, but did not exceed 35 mAh/g. Capacity retention after ca. 100 cycles was satisfactory. Further optimization of the residual-glass-to-nanocrystallite volume ratio would be desirable.

This paper presents a versatile model for generating fractal complex networks that closely mirror the properties of real-world systems. By combining features of reverse renormalization and evolving network models, the proposed approach introduces several tunable parameters, offering exceptional flexibility in capturing the diverse topologies and scaling behaviors found in both natural and man-made networks. The model effectively replicates their key characteristics such as fractal dimensions, power-law degree distributions, and scale-invariant properties of hierarchically nested boxes. Unlike traditional deterministic models, it incorporates stochasticity into the network growth process, overcoming limitations like discontinuities in degree distributions and rigid size constraints. The model's applicability is demonstrated through its ability to reproduce the structural features of real-world fractal networks, including the Internet, the World Wide Web, and co-authorship networks.

We investigate the full temporal evolution of epidemic outbreaks in complex networks, focusing on the susceptible-infected (SI) model of disease transmission. Combining theoretical analysis with large-scale numerical simulations, we uncover two universal patterns of epidemic growth, determined by the structure of the underlying network. In small-world networks, the prevalence follows a Gompertz-like curve, while in fractal networks it evolves according to Avrami-type dynamics, typical of spatially constrained systems. These regimes define distinct universality classes that remain robust across arbitrary transmission rates. Notably, our approach provides explicit analytical formulas for the global epidemic prevalence and class-specific scaling relations capturing its dependence on the transmission rate. We show that the commonly assumed early exponential growth occurs only in small-world networks, where it corresponds to the short-time approximation of the Gompertz function. In contrast, this exponential phase is entirely absent in fractal networks, where spreading is markedly slower and governed by different mechanisms. Our approach clarifies the structural origins of these contrasting behaviors and offers a unified framework for understanding epidemic dynamics across diverse network topologies.