A formula φ is called n-provable in a formal arithmetical theory S if φ is provable in S together with all true arithmetical ${{\rm{\Pi }}_n}$-sentences taken as additional axioms. While in general the set of all n-provable formulas, for a fixed $n > 0$, is not recursively enumerable, the set of formulas φ whose n-provability is provable in a given r.e. metatheory T is r.e. This set is deductively closed and will be, in general, an extension of S. We prove that these theories can be naturally axiomatized in terms of progressions of iterated local reflection principles. In particular, the set of provably 1-provable sentences of Peano arithmetic $PA$ can be axiomatized by ${\varepsilon _0}$ times iterated local reflection schema over $PA$. Our characterizations yield additional information on the proof-theoretic strength of these theories (w.r.t. various measures of it) and on their axiomatizability. We also study the question of speed-up of proofs and show that in some cases a proof of n-provability of a sentence can be much shorter than its proof from iterated reflection principles.

Scanning electron microscopy (SEM) of nanoscale objects in dry and fully hydrated conditions at different temperatures is of critical importance in revealing details of their interactions with an ambient environment. Currently available WETSEM capsules are equipped with thin electron-transparent membranes and allow imaging of samples at atmospheric pressure, but do not provide temperature control over the sample. Here, we developed and tested a thermoelectric cooling/heating setup for WETSEM capsules to allow ambient pressure in situ SEM studies with a temperature range between −15 and 100°C in gaseous, liquid, and frozen conditions. The design of the setup also allows for correlation of the SEM with optical microscopy and spectroscopy. As a demonstration of the possibilities of the developed approach, we performed real-time in situ microscopy studies of water condensation on a surface of Morpho sulkowskyi butterfly wing scales. We observed that initial water nucleation takes place on top of the scale ridges. These results confirmed earlier discovery of a preexisting polarity gradient of the ridges of Morpho butterflies. Our developed thermoelectric cooling/heating setup for environmental capsules meets the diverse needs for in situ nanocharacterization in material science, catalysis, microelectronics, chemistry, and biology.

We performed group-theoretical analysis of the symmetry relationships between lattice structures of R, M1, M2, and T phases of vanadium dioxide in the frameworks of the general Ginzburg-Landau phase transition theory. The analysis leads to a conclusion that the competition between the lower-symmetry phases M1, M2, and T in the metal-insulator transition is pure symmetry driven, since all the three phases correspond to different directions of the same multi-component structural order parameter. Therefore, the lower-symmetry phases can be stabilized in respect to each other by small perturbations such as doping or stress.