On the Fourth order Schr\"odinger equation in four dimensions: dispersive estimates and zero energy resonances

We study the fourth order Schr\"odinger operator $H=(-\Delta)^2+V$ for a decaying potential $V$ in four dimensions. In particular, we show that the $t^{-1}$ decay rate holds in the $L^1\to L^\infty$ setting if zero energy is regular. Furthermore, if the threshold energies are regular then a faster decay rate of $t^{-1}(\log t)^{-2}$ is attained for large $t$, at the cost of logarithmic spatial weights. Zero is not regular for the free equation, hence the free evolution does not satisfy this bound due to the presence of a resonance at the zero energy. We provide a full classification of the different types of zero energy resonances and study the effect of each type on the time decay in the dispersive bounds.Comment: Revised according to referee suggestions. To appear in J. Differential Equation

Dispersive estimates for massive Dirac operators in dimension two

We study the massive two dimensional Dirac operator with an electric potential. In particular, we show that the $t^{-1}$ decay rate holds in the $L^1\to L^\infty$ setting if the threshold energies are regular. We also show these bounds hold in the presence of s-wave resonances at the threshold. We further show that, if the threshold energies are regular that a faster decay rate of $t^{-1}(\log t)^{-2}$ is attained for large $t$, at the cost of logarithmic spatial weights. The free Dirac equation does not satisfy this bound due to the s-wave resonances at the threshold energies.Comment: 40 page

Scoping studies to establish the capability and utility of a real-time bioaerosol sensor to characterise emissions from environmental sources

A novel dual excitation wavelength based bioaerosol sensor with multiple fluorescence bands called Spectral Intensity Bioaerosol Sensor (SIBS) has been assessed across five contrasting outdoor environments. The mean concentrations of total and fluorescent particles across the sites were highly variable being the highest at the agricultural farm (2.6 cm−3 and 0.48 cm−3, respectively) and the composting site (2.32 cm−3 and 0.46 cm−3, respectively) and the lowest at the dairy farm (1.03 cm−3 and 0.24 cm−3, respectively) and the sewage treatment works (1.03 cm−3 and 0.25 cm−3, respectively). In contrast, the number-weighted fluorescent fraction was lowest at the agricultural site (0.18) in comparison to the other sites indicating high variability in nature and magnitude of emissions from environmental sources. The fluorescence emissions data demonstrated that the spectra at different sites were multimodal with intensity differences largely at wavelengths located in secondary emission peaks for λex 280 and λex 370. This finding suggests differences in the molecular composition of emissions at these sites which can help to identify distinct fluorescence signature of different environmental sources. Overall this study demonstrated that SIBS provides additional spectral information compared to existing instruments and capability to resolve spectrally integrated signals from relevant biological fluorophores could improve selectivity and thus enhance discrimination and classification strategies for real-time characterisation of bioaerosols from environmental sources. However, detailed lab-based measurements in conjunction with real-world studies and improved numerical methods are required to optimise and validate these highly resolved spectral signatures with respect to the diverse atmospherically relevant biological fluorophores