Symmetry

In ultrafast science, the strong-field approximation (SFA) provides a powerful framework to describe high-order harmonic generation (HHG) and related phenomena. Meanwhile, within the current ab initio theoretical framework, the use of nonlocal potentials in calculating multi-electron molecular wave functions is almost unavoidable. We find that when such wave functions are directly applied to compute transition dipole moments for correcting SFA, it introduces a fundamental gauge transformation problem. Specifically, the nonlocal potential contributes an additional gauge-dependent phase function to the dipole operator, which directly modifies the phase of the transition dipole. As a consequence, the saddle-point equations acquire an entirely different structure compared to the standard SFA, leading to a splitting of the conventional short and long classical trajectories in HHG into multiple distinct quantum trajectories. Here, ‘‘complex molecules’’ refers to multi-center molecular systems whose nonlocal electronic structure leads to gauge-dependent strong-field responses. Our analysis highlights that the validity of gauge in-variation cannot be assumed universally in SFA framework. Our approach combines the molecular strong-field approximation with gauge transformation analysis, incorporating nonlocal pseudopotentials, saddle-point equations, and multi-center recombination effects.

The simulation of seismic wave attenuation and dispersion in a fractured medium and the analysis of the influencing factors have an important guiding role for fracture detection and characterization. In this paper, for the fractured medium saturated with fluid, the finite element numerical simulation method of the Lamé–Navier and Navier–Stokes equations is investigated and compared with the numerical simulation method based on Biot’s equation. Biot’s method is more suitable for simulating fractured media at the mesoscopic scale, whereas for microscopic media, the Lamé–Navier and Navier–Stokes equations demonstrate distinct advantages. Meanwhile, the numerical simulation method is employed to analyze the influencing factors of connectivity of symmetrical fractures, effective compression length of seismic waves, and fluid viscosity. This analysis further elucidates the mechanisms and change characteristics of seismic wave attenuation and dispersion, providing theoretical guidance for the detection of fractures and fluids.

In this work, we introduce a new four-parameter distribution, called the integrated linear–Weibull (ILW) model, constructed by embedding a dynamic linear component within the Weibull framework. The ILW distribution is capable of capturing a wide variety of symmetric and asymmetric density shapes and accommodates diverse failure-rate behaviors. We derive several of its key mathematical and statistical properties, including moments, extropy, cumulative residual entropy, order statistics, and their asymptotic forms. The mean residual life function and its reciprocal relationship with the failure rate are also obtained. An algorithm for generating random samples from the ILW distribution is provided, and model identifiability is examined numerically through the Kullback–Leibler divergence. Parameter estimation is carried out via maximum likelihood, and a simulation study is conducted to assess the accuracy of the estimators; the results show improvements in estimator performance as sample size increases. Finally, three real datasets involving failure-time observations and one describing hydrological and epidemiological data are analyzed to demonstrate the practical usefulness of the ILW model. In these applications, the proposed model exhibits competitive or superior performance relative to several existing lifetime distributions based on standard model selection criteria and goodness-of-fit measures.