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We show the refinement of the prescription for the geometric transition in the refined topological string theory and, as its application, discuss a possibility to describe $qq$-characters from the string theory point of view. Though the suggested way to operate the refined geometric transition has passed through several checks, it is additionally found in this paper that the presence of the preferred direction brings a nontrivial effect. We provide the modified formula involving this point. We then apply our prescription of the refined geometric transition to proposing the stringy description of doubly quantized Seiberg--Witten curves called $qq$-characters in certain cases., 44 pages, 11 figures; v2: references corrected, text corrected, published in JHEP

We compute the exact partition function on the branched two-sphere by the localization technique. It is found that it does not depend on a branching parameter q, which means that supersymmetric R\'enyi entropy defined by utilizing it is equivalent to the usual entanglement entropy. We also provide the interpretation of the conical singularities on the branched sphere as defects sit on the poles of the nonsingular two-sphere., Comment: 27 pages; v2: typos and discussions revised, and reference added; v3: typos corrected and published in JHEP

We study 3d mirror symmetry with loop operators, Wilson loop and Vortex loop, and multi-flavor mirror symmetry through utilizing the $$ \mathrm{\mathbb{R}}{\mathrm{\mathbb{P}}}^2\times {\mathbb{S}}^1 $$ index formula. The key identity which makes the above description work well is the mod 2 version of the Fourier analysis, and we study such structure, the S-operation in the context of a $$ \mathrm{S}\mathrm{L}\left(2,\mathrm{\mathbb{Z}}\right) $$ action on 3d SCFTs. We observed that two types of the parity conditions basically associated with gauge symmetries which we call $$ \mathcal{P} $$ -type and $$ \mathcal{C}\mathcal{P} $$ -type are interchanged under mirror symmetry. We will also comment on the T-operation.

Multi-step precipitation separation system was developed by using aqueous mixtures of some thermosensitive polymers. The following three polymers were used here; poly(N-n-propylacrylamide), poly(N-isopropylacrylamide), and poly(N-isopropylmethacrylamide). A mixture of the three polymers showed three endothermic peaks, and the peak top temperatures were almost consistent with that of the each polymer solution. The polymers were purified by thermal precipitation to obtain fractions which can respond in narrow temperature ranges prior to use. In the case of the precipitation separation of two polymers mixtures, purities of the obtained precipitate and supernatant fractions became high comparing with the case in which the unpurified polymers were used. Parts of the polymers which were not the precipitation targets were also precipitated by the separation procedures. This was caused not only by insolubilization of the non-targeted polymers due to their phase transitions but also by their non-specific entanglement with the targeted polymers. The purities of the fractions also improved when the difference of the phase transition temperature between two polymers was large enough to avoid the coprecipitation. In the case of the precipitation separation of mixtures of the three polymers, purities of each fraction also improved when the purified polymers were used.