We introduce the notion of completed F-crystals on the absolute prismatic site of a smooth p-adic formal scheme. We define a functor from the category of completed prismatic F-crystals to that of crystalline étale Zp-local systems on the generic fiber of the formal scheme and show that it gives an equivalence of categories. This generalizes the work of Bhatt and Scholze, which treats the case of a complete discrete valuation ring with perfect residue field.

We give a new construction of (φ,Ĝ )-modules using the theory of prisms developed by Bhatt and Scholze. As an application, we give a new proof about the equivalence between the category of prismatic F-crystals in finite locally free Δ-modules over (K)Δ and the category of lattices in crystalline representations of GK, where K is a complete discretely valued field of mixed characteristic with perfect residue field. We also generalize this result to semi-stable representations using the absolute logarithmic prismatic site defined by Koshikawa.

Let R be a non-discrete rank one valuation ring of characteristic p and let E be any discrete valuation ring, we prove the ring of E-Witt vectors over R has uncountable Krull dimension without assuming the axiom of existence of prime ideals for general commutative unitary rings.

Let K be a discrete valuation field, we combine the construction of Fargues-Fontaine of GK-equivariant modifications of vector bundles over the Fargues-Fontaine curve XFF using weakly admissible filtered (φ,N,GK)-modules over K, with Scholze and Fargues' theorems that relate modifications of vector bundles over the Fargues-Fontaine curve with mixed characteristic shtukas and Breuil-Kisin-Fargues modules. We give a characterization of Breuil-Kisin-Fargues modules with semilinear GK-actions that produced in this way and compare those Breuil-Kisin-Fargues modules with Kisin modules.

We analyze the power spectra and structure functions (SFs) of the temperature and radial velocity fields, calculated in the radial and azimuthal directions, in annular centrifugal Rayleigh–Bénard convection (ACRBC) for Rayleigh number Ra ∈[108,1011], Prandtl number Pr = 10.7, and inverse Rossby number Ro−1=16 using the spatial data obtained by quasi-two-dimensional direct numerical simulation. Bolgiano and Obukhov-like (BO59-like) scalings for the energy spectrum in both the azimuthal and radial directions and thermal spectrum in the azimuthal direction are observed. The range of BO59-like scaling becomes wider as Ra increases. At Ra=1011, it is found that BO59-like scaling Eu(kr)∼kr−11/5 spans nearly two decades for the energy spectrum calculated in the radial direction. Power-law fittings in the range larger than the Bolgiano scales, the scaling exponents of transverse and longitudinal velocity SFs vs the order coincide with the theoretical prediction of BO59 scaling ζ u p =3p/5 basically. The second-order temperature SFs exhibit a gradual transition from the Obukhov–Corrsin behavior at scales smaller than the Bolgiano scales to the BO59 behavior at scales larger than the Bolgiano scales. The slopes from the third to sixth-order temperature SFs are similar, which is similar to classical Rayleigh–Bénard convection and Rayleigh–Taylor turbulence. The probability density functions (p.d.f.) of temperature fluctuations δT/σT reveal the cold plumes are strong and the p.d.f. in different regions at high Ra are similar. The stronger turbulent-mixing and larger centrifugal buoyancy in ACRBC may result in the BO59-like scaling.