We describe an instrument that allows the rapid measurement of fluorescence lifetime-resolved images of leaves as well as sub-cellular structures of intact plants or single cells of algae. Lifetime and intensity fluorescence images can be acquired and displayed in real time (up to 55 lifetime-resolved images per s). Our imaging technique therefore allows rapid measurements that are necessary to determine the fluorescence lifetimes at the maximum (P level) fluorescence following initial illumination during the chlorophyll (Chl) a fluorescence transient (induction) in photosynthetic organisms. We demonstrate the application of this new instrument and methodology to measurements of: (1) Arabidopsis thaliana leaves showing the effect of dehydration on the fluorescence lifetime images; (2) Zea mays leaves showing differences in the fluorescence lifetimes due to differences in the bundle sheath cells (having a higher amount of low yield photosystem 1) and the mesophyll cells (having a higher amount of high yield photosystem 2); and (3) single cells of wild type Chlamydomonas reinhardtii and its non-photochemical quenching mutant NPQ2 (where the conversion of zeaxanthin to violaxanthin is blocked), with NPQ2 showing lowered lifetime of Chl a fluorescence. In addition to the lifetime differences referred to in (1) and (2), structural dependent heterogeneities in the fluorescence lifetimes were generally observed when imaging mesophyll cells in leaves. and O. Holub ... [et al.].
The second order linear difference equation (1) ∆(rk∆xk) + ckxk+1 = 0, where rk ≠ 0 and k ∈ ℤ , is considered as a special type of symplectic systems. The concept of the phase for symplectic systems is introduced as the discrete analogy of the Borůvka concept of the phase for second order linear differential equations. Oscillation and nonoscillation of (1) and of symplectic systems are investigated in connection with phases and trigonometric systems. Some applications to summation of number series are given, too.