Fe-based amorphous and nanocrystalline alloys, such as FINEMET and its improved variants, are highly valued as green energy-saving materials due to their unique magnetic properties, including high permeability, low coercivity, and near-zero saturation magnetostriction. These characteristics have enabled their extensive use in power electronics and information technology. However, the full potential of these alloys remains unfulfilled due to insufficient understanding of their stress sensitivity. This study focuses on the development history, heat treatment, annealing processes, chemical composition, and underlying mechanisms of Fe-based amorphous and nanocrystalline alloys, aiming to provide insights into stress-induced magnetic anisotropy and guide the development of greener and more efficient soft magnetic materials.

The dissipation phenomenon in the heavy-ion reaction at incident energy near the Fermi energy is studied by simulating the reaction 129Xe+129Sn with the isospin-dependent quantum molecular dynamics model. The calculations involving a proper prescription of implementing the Pauli exclusion principle show that the isotropy ratio measured by free protons emitted in the reaction at energy slightly higher than the Fermi energy is in agreement with the experimental data recently released by the INDRA collaboration. A feasible value of the Pauli-blocking factor is estimated by comparing the theoretical results with the experimental data for the energy range considered here.

The 2p photoionization of sodium atoms from the initial states 2p ⁶3p _1/2,3/2 have been investigated, with special emphasis placed on the relaxation and correlation effects. To understand the details of the spectra, a series of ab initio calculations based on the multi-configuration Dirac–Fock method have been performed. Theoretical predictions have been compared with the high-resolution spectra measured using a synchrotron radiation source, and suitable agreement between experiment and theory has enabled the identification of fine-structure spectra. The prominent influence of the relaxation and correlation effects on the photoelectron spectra has been analysed according to the detailed calculations.

The Collatz map is a simple deterministic transformation whose orbit structure remains highly nontrivial. A recent direction-phase decomposition partitions each orbit into upward and downward steps, and numerical observations indicate that the number of upward phases, $N_{\uparrow }$, follows an approximate Gamma distribution. In this work, we provide a mechanistic explanation for this statistical regularity by modeling the occurrence of upward phases in the odd-compressed, or Syracuse, version of the Collatz map as a homogeneous Poisson process. From the mean-field logarithmic balance and the geometric distribution of 2-adic valuations, we derive closed-form expressions for the Gamma parameters: the scale parameter $\theta =2/{(2-{\log }_{2}3)}^2\approx 11.61$ is constant, whereas the shape parameter K grows logarithmically with the maximal initial value $X_0=2L+1$. We also analyze the closure conditions for periodic orbits, showing that nontrivial cycles are severely constrained, which supports the plausibility of the statistical framework. Numerical validation for L ranging from ${10}^5$ to ${10}^{15}$ confirms the theory with relative errors below $3\%$, and a bias-corrected mean estimate reduces the error to ${10}^{-3}$–${10}^{-2}\%$. These results establish a quantitative link between the arithmetic properties of the Collatz map and Gamma-type statistics, and suggest possible extensions to generalized Collatz-type problems.