In a completely systematic and geometric way, we derive maximal and halfmaximal supersymmetric gauged double field theories in lower than ten dimensions. To this end, we apply a simple twisting ansatz to the D = 10 ungauged maximal and halfmaximal supersymmetric double field theories constructed previously within the so-called semi-covariant formalism. The twisting ansatz may not satisfy the section condition. Nonetheless, all the features of the semi-covariant formalism, including its complete covariantizability, are still valid after the twist under alternative consistency conditions. The twist allows gaugings as supersymmetry preserving deformations of the D = 10 untwisted theories after Scherk-Schwarz-type dimensional reductions. The maximal supersymmetric twist requires an extra condition to ensure both the Ramond-Ramond gauge symmetry and the 32 supersymmetries unbroken.

Finite systems may undergo first or second order phase transitions under not isovolumetric but isobaric condition. The 'analyticity' of a finite-system partition function has been argued to imply universal values for isobaric critical exponents, α P , β P and γ P . Here we test this prediction by analyzing NIST REFPROP data for twenty major molecules, including H 2 O, CO 2 , O 2 , etc. We report they are consistent with the prediction for temperature range, 10 −5 < |T /T c − 1| < 10 −3 . For each molecule, there appears to exist a characteristic natural number, n = 2, 3, 4, 5, 6, which determines all the critical exponents for T < T c as α P = γ P = n n+1 and β P = δ −1 = 1 n+1 . For the opposite T > T c , all the fluids seem to indicate the universal value of n = 2

Evading the Mermin-Wagner-Hohenberg no-go theorem and revisiting with rigor the ideal Bose gas confined in a square box, we explore a discrete phase transition in two spatial dimensions. Through both analytic and numerical methods, we verify that thermodynamic instability emerges if the number of particles is sufficiently yet finitely large: specifically, ⩾ N 35131. The instability implies that the isobar of the gas zigzags on the temperature-volume plane, featuring supercooling and superheating phenomena. The Bose-Einstein condensation can then persist from absolute zero to the superheating temperature. Without necessarily taking the large N limit, under a constant pressure condition, the condensation takes place discretely both in the momentum and in the position spaces. Our result is applicable to a harmonic trap. We assert that experimentally observed Bose-Einstein condensations of harmonically trapped atomic gases are a first-order phase transition that involves a discrete change of the density at the center of the trap.