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H2 and CO are mostly regarded as the signature products before the thermal runaway of lithium batteries. In fact, most small‐molecule gases result from the electrolyte decomposition inside the lithium battery under high temperature. The main component of electrolyte, dimethyl carbonate (DMC) can spill out of the case much earlier than H2 and CO. Herein, it is studied that the gas production of a lithium battery before its thermal runaway, and verified that gaseous DMC is a much earlier marker to warn thermal runaway. To solve the lack of highly DMC‐sensitive sensors, a novel semiconductor gas sensor is fabricated by Bi2O3 nanosheets. It performs high sensitivity and selectivity toward DMC with ultra‐low limit of detection (50 ppb) and high selectivity (>24 times). Notably, in comparison with temperature measurement and other commercial gas sensors, the as‐prepared Bi2O3 sensor detecting gaseous DMC can provide an early warning over 15 min before a thermal runaway happened. So, this Bi2O3 sensor has a great potential for practical application on warning thermal runaway. The in situ infrared spectroscopy and the in situ Raman spectroscopy are employed to investigate the nature behind the outstanding performances, which is ascribed to the direct interaction between electrolyte molecules and Bi2O3nanosheets. [ABSTRACT FROM AUTHOR]

For nonautonomous linear difference equations, we introduce the notion of the so-called nonuniform dichotomy spectrum and prove a spectral theorem. As an application of the spectral theorem, we prove a reducibility result.

When the vorticity is monotone with depth, we present a variational formulation for steady periodic water waves of the equatorial flow in the f-plane approximation, and show that the governing equations for this motion can be obtained by studying variations of a suitable energy functional 𝓗 in terms of the stream function and the thermocline. We also compute the second variation of the constrained energy functional, which is related to the linear stability of steady water waves.

We present here exact solutions to the equations of geophysical fluid dynamics that depict inviscid flows moving in the azimuthal direction on a circular path, around the globe, and which admit a velocity profile below the surface and along it. These features render this model suitable for the description of the Antarctic circumpolar current (ACC). The governing equations we work with–taken to be the Euler equations written in spherical coordinates–also incorporate forcing terms which are generally regarded as means that ensure the general balance of the ACC. Our approach allows for a variable density (depending on the depth and latitude) of discontinuous type which divides the water domain into two layers. Thus, the discontinuity gives rise to an interface. The velocity in both layers and the pressure in the lower layer are determined explicitly, while the pressure in the upper layer depends on the free surface and the interface. Functional analytical techniques render (uniquely) the surface and interface-defining functions in an implicit way. We conclude our discussion by deriving relations between the monotonicity of the surface pressure and the monotonicity of the surface distortion that concur with the physical expectations. A regularity result concerning the interface is also derived. [ABSTRACT FROM AUTHOR]