On a Symmetrization of Diffusion Processes

The latter author, together with collaborators, proposed a numerical scheme to calculate the price of barrier options. The scheme is based on a symmetrization of diffusion process. The present paper aims to give a mathematical credit to the use of the numerical scheme for Heston or SABR type stochastic volatility models. This will be done by showing a fairly general result on the symmetrization (in multi-dimension/multi-reflections). Further applications (to time-inhomogeneous diffusions/ to time dependent boundaries/to curved boundaries) are also discussed

Hydrodynamic Simulations of Merging Galaxy Clusters: Non-Equilibrium Ionization State and Two-Temperature Structure

We investigate a non-equilibrium ionization state and an electron-ion two-temperature structure of the intracluster medium (ICM) in merging galaxy clusters using a series of N-body and hydrodynamic simulations. Mergers with various sets of mass ratios and impact parameters are systematically investigated, and it is found that, in most cases, ICM significantly departs from the ionization equilibrium state at the shock layers with a Mach number of ~1.5-2.0 in the outskirts of the clusters, and the shock layers with a Mach number of ~2-4 in front of the ICM cores. Accordingly, the intensity ratio between Fe xxv and Fe xxvi K alpha line emissions is significantly altered from that in the ionization equilibrium state. If the effect of the two-temperature structure of ICM is incorporated, the electron temperature is ~10-20 % and ~30-50 % lower than the mean temperature of ICM at the shock layers in the outskirts and in front of the ICM cores, respectively, and the deviation from the ionization equilibrium state becomes larger. We also address the dependence of the intensity ratio on the viewing angle with respect to the merging plane.Comment: 11 pages, 10 figures. Submitted to PASJ; Accepted for publication in PAS

Blowup and Scattering problems for the Nonlinear Schr\"odinger equations

We consider $L^{2}$-supercritical and $H^{1}$-subcritical focusing nonlinear Schr\"odinger equations. We introduce a subset $PW$ of $H^{1}(\mathbb{R}^{d})$ for $d\ge 1$, and investigate behavior of the solutions with initial data in this set. For this end, we divide $PW$ into two disjoint components $PW_{+}$ and $PW_{-}$. Then, it turns out that any solution starting from a datum in $PW_{+}$ behaves asymptotically free, and solution starting from a datum in $PW_{-}$ blows up or grows up, from which we find that the ground state has two unstable directions. We also investigate some properties of generic global and blowup solutions