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Let $X$ and $Y$ be independent variance-gamma random variables with zero location parameter; then the exact probability density function of the ratio $X/Y$ is derived. Some basic distributional properties are also derived, including identification of parameter regimes under which the density is bounded, asymptotic approximations of tail probabilities, and fractional moments; in particular, we see that the mean is undefined. In the case that $X$ and $Y$ are independent symmetric variance-gamma random variables, an exact formula is also given for the cumulative distribution function of the ratio $X/Y$.

Spreading dynamics of liquid droplets impacting onto powder bed are experimentally studied using high-speed photography. Dimensionless numbers—We, Re, the modified We* and Re∗ corrected by substrate deformation—are used to analyze the impact behaviors of droplets. The spreading time and the maximum spreading factor are further analyzed. The spreading time is accurately described by a universal scaling law that is obtained from the modified dimensionless time vs. the effective Weber number (We∗), and the maximum spreading factor is found to follow the modified classic scaling law βmax = f(We*, Re*).

In recent years, with the popularization of intelligent auxiliary driving technology, wireless charging technology has garnered widespread attention due to its advantages in automation. However, there are still many issues to be resolved before the widespread adoption of wireless charging technology, with a key issue being the interoperability of coupling structures. To achieve adaptive interoperability, this paper proposes a wireless charging system based on dual generalized double-D (DD) coils. The system introduces a receiving structure composed of two generalized DD coils, which can naturally decouple the two coils. At the same time, the system topology employs a series connection of half-bridge rectifiers to enhance the system’s equivalent mutual inductance. Mathematical models and simulation verifications are provided. The proposed approach has been validated through simulations, showing that the system can automatically achieve interoperability with unipolar coils, DD coils, and quadrupolar coils.