We consider the $\mathbb {T}^{4}$ cubic nonlinear Schrödinger equation (NLS), which is energy-critical. We study the unconditional uniqueness of solutions to the NLS via the cubic Gross–Pitaevskii hierarchy, an uncommon method for NLS analysis which is being explored [24, 35] and does not require the existence of a solution in Strichartz-type spaces. We prove U-V multilinear estimates to replace the previously used Sobolev multilinear estimates. To incorporate the weaker estimates, we work out new combinatorics from scratch and compute, for the first time, the time integration limits, in the recombined Duhamel–Born expansion. The new combinatorics and the U-V estimates then seamlessly conclude the $H^{1}$ unconditional uniqueness for the NLS under the infinite-hierarchy framework. This work establishes a unified scheme to prove $H^{1}$ uniqueness for the $ \mathbb {R}^{3}/\mathbb {R}^{4}/\mathbb {T}^{3}/\mathbb {T}^{4}$ energy-critical Gross–Pitaevskii hierarchies and thus the corresponding NLS.

The failure mechanism of lead-free solder interconnections of chip scale package–sized Ball Grid Array (BGA) component boards under thermal cycling was studied by employing cross-polarized light microscopy, scanning electronic microscopy, electron backscatter diffraction, and nanoindentation. It was determined that the critical solder interconnections were located underneath the chip corners, instead of the corner most interconnections of the package, and the highest strains and stresses were concentrated at the outer neck regions on the component side of the interconnections. Observations of the failure modes were in good agreement with the finite element results. The failure of the interconnections was associated with changes of microstructures by recrystallization in the strain concentration regions of the solder interconnections. Coarsening of intermetallic particles and the disappearance of the boundaries between the primary Sn cells were observed in both cases. The nanoindentation results showed lower hardness of the recrystallized grains compared with the non-recrystallized regions of the same interconnection. The results show that failure modes are dependent on the localized microstructural changes in the strain concentration regions of the interconnections and the crack paths follow the networks of grain boundaries produced by recrystallization.