In this paper, we consider an extended concept of invariant for polynomial dynamical systems (PDSs) with domain and initial condition, and establish a sound and complete criterion for checking semi-algebraic invariants (SAIs) for such PDSs. The main idea is encoding relevant dynamical properties as conditions on the high order Lie derivatives of polynomials occurring in the SAI. A direct consequence of this criterion is a relatively complete method of SAI generation based on template assumption and semi-algebraic constraint solving. Relative completeness means if there is an SAI in the form of a predefined template, then our method can indeed find one.

A novel photocatalysis and gas sensing material was synthesized by decorating Au nanoparticles on tungsten trioxide nanorods. Tungsten trioxide nanorods were prepared through the ion-exchange method combined with hydrothermal treatment and further modified with Au nanoparticles (Au NPs). After Au NPs decorated on the surface of WO3 nanorods (WO3 NRs), the reducing gas (hydrogen, methanol, ethanol, etc.) sensing properties and the photocatalytic performance of rhodamine B (RhB) were all greatly improved. Au NP modified WO3 nanorods (Au NP@WO3 NRs) exhibit not only larger response (H2 50 ppm, recovery time lower than 10 s) and better selectivity (Ra/Rg = 6.6) for H2 gas detection than pure WO3 NRs but also high photocatalytic properties for the absolute degradation of RhB under simulated sunlight irradiation for 120 min.

A facile solvothermal route to synthesize S,N co‐doped graphene quantum dots (S,N‐GQDs) with unique optical properties is demonstrated. Three absorption bands are observed at 338, 467, and 557 nm, which is different from any previous reports. The photoluminescent spectra display emissions in three primary colors that are independent of the excitation wavelength, within the excitation wavelength ranges of 340–420 nm, 460–540 nm, and 560–620 nm. The PL excitation spectra indicate that each emission is related to a single excitation band. It is proposed that three independent luminescent centers coexist in S,N‐GQDs because the doping with S and N may change the chemical environment of the GQDs. However, energy‐transfer processes usually do not occur among the independent luminescent centers under different wavelength light excitation. Heteroatom‐doping of GQDs provides an attractive means of effectively tuning their optical properties for the purpose of exploiting new applications in visible‐light photocatalytic and bioimaging. S,N‐GQDs/TiO2 composites exhibit better hydrogen production activities under visible light (λ > 420 nm) than commercial TiO2 (P25), owing to the presence of characteristic absorption bands in the visible region. Furthermore, the S,N‐GQDs have a pronounced biocompatibility and bioimaging ability under long‐wavelengths excitation for live A549 cells.