GEMS Embeddings of Schwarzschild and RN Black Holes in Painlevé-Gullstrand Spacetimes

Making use of the higher dimensional global embedding Minkowski spacetime (GEMS), we embed (3 + 1)-dimensional Schwarzschild and Reissner-Nordström (RN) black holes written by the Painlevé-Gullstrand (PG) spacetimes, which have off-diagonal components in metrics, into (5 + 1)- and (5 + 2)-dimensional flat ones, respectively. As a result, we have shown the equivalence of the GEMS embeddings of the spacetimes with the diagonal and off-diagonal terms in metrics. Moreover, with the aid of their geodesic equations satisfying various boundary conditions in the flat embedded spacetimes, we directly obtain freely falling temperatures. We also show that freely falling temperatures in the PG spacetimes are well-defined beyond the event horizons, while they are equivalent to the Hawking temperatures, which are obtained in the original curved ones in the ranges between the horizon and the infinity. These will be helpful to study GEMS embeddings of more realistic Kerr, or rotating BTZ black holes.

Asymptotic Self-Similarity of Minimizers and Local Bounds in a Model of Shape-Memory Alloys

We prove that microstructures in shape-memory alloys have a self-similar refinement pattern close to austenite-martensite interfaces, working within the scalar Kohn-Müller model. The latter is based on nonlinear elasticity and includes a singular perturbation representing the energy of the interfaces between martensitic variants. Our results include the case of low-hysteresis materials in which one variant has a small volume fraction. Precisely, we prove asymptotic self-similarity in the sense of strong convergence of blow-ups around points at the austenite-martensite interface. Key ingredients in the proof are pointwise estimates and local energy bounds. This generalizes previous results by one of us to various boundary conditions, arbitrary rectangular domains, and arbitrary volume fractions of the martensitic variants, including the regime in which the energy scales as $\varepsilon ^{2/3}$ ε 2 / 3 as well as the one where the energy scales as $\varepsilon ^{1/2}$ ε 1 / 2 .

A Comparative Study of Atmospheric Chemistry with VULCAN

We present an update of the open-source photochemical kinetics code VULCAN to include C–H–N–O–S networks and photochemistry. The additional new features are advection transport, condensation, various boundary conditions, and temperature-dependent UV cross sections. First, we validate our photochemical model for hot Jupiter atmospheres by performing an intercomparison of HD 189733b models between Moses et al., Venot et al., and VULCAN, to diagnose possible sources of discrepancy. Second, we set up a model of Jupiter extending from the deep troposphere to upper stratosphere to verify the kinetics for low temperature. Our model reproduces hydrocarbons consistent with observations, and the condensation scheme successfully predicts the locations of water and ammonia ice clouds. We show that vertical advection can regulate the local ammonia distribution in the deep atmosphere. Third, we validate the model for oxidizing atmospheres by simulating Earth and find agreement with observations. Last, VULCAN is applied to four representative cases of extrasolar giant planets: WASP-33b, HD 189733b, GJ 436b, and 51 Eridani b. We look into the effects of the C/O ratio and chemistry of titanium/vanadium species for WASP-33b, we revisit HD 189733b for the effects of sulfur and carbon condensation, the effects of internal heating and vertical mixing (K zz) are explored for GJ 436b, and we test updated planetary properties for 51 Eridani b with S8 condensates. We find that sulfur can couple to carbon or nitrogen and impact other species, such as hydrogen, methane, and ammonia. The observable features of the synthetic spectra and trends in the photochemical haze precursors are discussed for each case.

Ambarzumian-type Problems for Discrete Schrödinger Operators

We discuss the problem of unique determination of the finite free discrete Schrödinger operator from its spectrum, also known as the Ambarzumian problem, with various boundary conditions, namely any real constant boundary condition at zero and Floquet boundary conditions of any angle. Then we prove the following Ambarzumian-type mixed inverse spectral problem: diagonal entries except the first and second ones and a set of two consecutive eigenvalues uniquely determine the finite free discrete Schrödinger operator.