Hyperbolic Schwarz map for the hypergeometric differential equation

The Schwarz map of the hypergeometric differential equation is studied since the beginning of the last century. Its target is the complex projective line, the 2-sphere. This paper introduces the hyperbolic Schwarz map, whose target is the hyperbolic 3-space. This map can be considered to be a lifting to the 3-space of the Schwarz map. This paper studies the singularities of this map, and visualize its image when the monodromy group is a finite group or a typical Fuchsian group. General cases will be treated in a forthcoming paper.Comment: 16 pages, 8 figure

Derived Schwarz map of the hypergeometric differential equation and a parallel family of flat fronts

In the previous paper (math.CA/0609196) we defined a map, called the hyperbolic Schwarz map, from the one-dimensional projective space to the three-dimensional hyperbolic space by use of solutions of the hypergeometric differential equation, and thus obtained closed flat surfaces belonging to the class of flat fronts. We continue the study of such flat fronts in this paper. First, we introduce the notion of derived Schwarz maps of the hypergeometric differential equation and, second, we construct a parallel family of flat fronts connecting the classical Schwarz map and the derived Schwarz map.Comment: 15 pages, 12 figure

Quantum algorithm for collisionless Boltzmann simulation of self-gravitating systems

The collisionless Boltzmann equation (CBE) is a fundamental equation that governs the dynamics of a broad range of astrophysical systems from space plasma to star clusters and galaxies. It is computationally expensive to integrate the CBE directly in a phase space, and thus the applications to realistic astrophysical problems have been limited so far. Recently, Todorova \& Steijl (2020) proposed an efficient quantum algorithm for solving the CBE with a significantly reduced computational complexity. We extend the method to perform quantum simulations that follow the evolution of self-gravitating systems. We first run a 1+1 dimensional test calculation of free streaming motion on 64$\times$64 grids using 13 simulated qubits and validate our method. We then perform simulations of Jeans collapse, and compare the result with analytic and linear theory calculations. We propose a direct method to generate initial conditions as well as a method to retrieve necessary information from a register of multiple qubits. Our simulation scheme achieves $\mathcal{O}(N_v^3)$ less computational complexity than the classical method, where $N_v$ is the number of discrete velocity grids per dimension. It will thus allow us to perform large-scale CBE simulations on future quantum computers.Comment: 10 pages, 9figure

Simultaneous Improvements in Performance and Durability of an Octahedral PtNix/C Electrocatalyst for Next-Generation Fuel Cells by Continuous, Compressive, and Concave Pt Skin Layers

Simultaneous improvements in oxygen reduction reaction (ORR) activity and long-term durability of Pt-based cathode catalysts are indispensable for the development of next-generation polymer electrolyte fuel cells but are still a major dilemma. We present a robust octahedral core–shell PtNix/C electrocatalyst with high ORR performance (mass activity and surface specific activity 6.8–16.9 and 20.3–24.0 times larger than those of Pt/C, respectively) and durability (negligible loss after 10000 accelerated durability test (ADT) cycles). The key factors of the robust octahedral nanostructure (core–shell Pt73Ni27/C) responsible for the remarkable activity and durability were found to be three continuous Pt skin layers with 2.0–3.6% compressive strain, concave facet arrangements (concave defects and high coordination), a symmetric Pt/Ni distribution, and a Pt67Ni33 intermetallic core, as found by STEM-EDS, in situ XAFS, XPS, etc. The robust core–shell Pt73Ni27/C was produced by the partial release of the stress, Pt/Ni rearrangement, and dimension reduction of an as-synthesized octahedral Pt50Ni50/C with 3.6–6.7% compressive Pt skin layers by Ni leaching during the activation process. The present results on the tailored synthesis of the PtNix structure and composition and the better control of the robust catalytic architecture renew the current knowledge and viewpoint for instability of octahedral PtNix/C samples to provide a new insight into the development of next-generation PEFC cathode catalysts

Equilibriums of extremely magnetized compact stars with force-free magnetotunnels

We present numerical solutions for stationary and axisymmetric equilibriums of compact stars associated with extremely strong magnetic fields. The interior of the compact stars is assumed to satisfy ideal magnetohydrodynamic (MHD) conditions, while in the region of negligible mass density the force-free conditions or electromagnetic vacuum are assumed. Solving all components of Einstein's equations, Maxwell's equations, ideal MHD equations, and force-free conditions, equilibriums of rotating compact stars associated with mixed poloidal and toroidal magnetic fields are obtained. It is found that in the extreme cases the strong mixed magnetic fields concentrating in a toroidal region near the equatorial surface expel the matter and form a force-free toroidal magnetotunnel. We also introduce a new differential rotation law for computing solutions associated with force-free magnetosphere, and present other extreme models without the magnetotunnel.Comment: 13 pages, 4 figure