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Abstract We show that R. Thompson’s group Tis a maximal subgroup of the group V. The argument provides examples of foundational calculations which arise when expressing elements ofVas products of transpositions of basic clopen sets in the Cantor space $$\mathfrak {C}$$ $C$. 

Abstract Terrestrial Gamma‐ray Flashes (TGFs) are ten‐to‐hundreds of microsecond bursts of gamma‐rays produced when electrons in strong electric fields in thunderclouds are accelerated to relativistic energies. Space instruments have observed TGFs with source photon brightness down to ∼10¹⁷–10¹⁶. Based on space and aircraft observations, TGFs have been considered rare phenomena produced in association with very few lightning discharges. Space observations associated with lightning ground observations in the radio band have indicated that there exists a population of dimmer TGFs. Here we show observations of TGFs from aircraft altitude that were not detected by a space instrument viewing the same area. The TGFs were found through Monte Carlo modeling to be associated with 10¹⁵–10¹²photons at source, which is several orders of magnitude below what can be seen from space. Our results suggest that there exists a significant population of TGFs that are too weak to be observed from space. 

Resonance stimulated Raman signal and line shape are evaluated analytically under common electronic/vibrational dephasing and exponential Raman/probe pulse, exp(−|t|/τ). Generally, the signal from a particular state includes contributions from higher and lower electronic states. Thus, with S0 → S1 actinic excitation, the Raman signal consists of 15 Feynman diagrams entering with different signs. The negative sign indicates vibrational coherences in S1 or higher Sn, whereas the positive sign reveals coherences in S0 or Sn via S1 → Sn → Sm (n < m) coupling. The signal complexity is in contrast to spontaneous Raman with its single diagram only. The results are applied to femtosecond stimulated Raman spectra of trans–trans, cis–trans (ct), and cis–cis (cc) 1,4-diphenyl-1,3-butadiene, the ct and cc being reported for the first time. Upon actinic excitation, the Stokes spectra show negative bands from S1 or Sn. When approaching higher resonances Sn → Sm, some Raman bands switch their sign from negative to positive, thus, indicating new coherences in Sn. The results are discussed, and the measured Raman spectra are compared to the computed quantum-chemical spectra.