The discrete-time quaternionic quantum walk and the second weighted zeta function on a graph

We define the quaternionic quantum walk on a finite graph and investigate its properties. This walk can be considered as a natural quaternionic extension of the Grover walk on a graph. We explain the way to obtain all the right eigenvalues of a quaternionic matrix and a notable property derived from the unitarity condition for the quaternionic quantum walk. Our main results determine all the right eigenvalues of the quaternionic quantum walk by using complex eigenvalues of the quaternionic weighted matrix which is easily derivable from the walk. Since our derivation is owing to a quaternionic generalization of the determinant expression of the second weighted zeta function, we explain the second weighted zeta function and the relationship between the walk and the second weighted zeta function.Comment: 15 page

First Evidence of a Retrograde Orbit of Transiting Exoplanet HAT-P-7b

We present the first evidence of a retrograde orbit of the transiting exoplanet HAT-P-7b. The discovery is based on a measurement of the Rossiter-McLaughlin effect with the Subaru HDS during a transit of HAT-P-7b, which occurred on UT 2008 May 30. Our best-fit model shows that the spin-orbit alignment angle of this planet is \lambda = -132.6 (+10.5, -16.3) degrees. The existence of such a retrograde planet have been predicted by recent planetary migration models considering planet-planet scattering processes or the Kozai migration. Our finding provides an important milestone that supports such dynamic migration theories.Comment: PASJ Letters, in press [13 pages

Stress analysis of quasi-orthotropic elastic plane

AbstractThe quasi-orthotropic elastic plane in which the characteristic roots of the fundamental differential equation for the orthotropic elastic plane are doubled is investigated. For the associated stress analysis, a rational mapping function is used and a closed-form solution is obtained. Therefore, the stress analysis is rigorous for the rational mapping function. The stress functions can be obtained without any integration. As a demonstration of the stress analysis, a half plane with an oblique edge crack is analyzed. Stress distributions and stress intensity factors are investigated. The relationships between the quasi-orthotropic elastic plane and isotropic plane with respect to the stress intensity factor are investigated. It is also determined that the stress intensity factors for the quasi-orthotropic elastic plane can be calculated from those of the isotropic elastic plane using a similar configuration and loading condition

Ronkin/Zeta Correspondence

The Ronkin function was defined by Ronkin in the consideration of the zeros of almost periodic function. Recently, this function has been used in various research fields in mathematics, physics and so on. Especially in mathematics, it has a closed connections with tropical geometry, amoebas, Newton polytopes and dimer models. On the other hand, we have been investigated a new class of zeta functions for various kinds of walks including quantum walks by a series of our previous work on Zeta Correspondence. The quantum walk is a quantum counterpart of the random walk. In this paper, we present a new relation between the Ronkin function and our zeta function for random walks and quantum walks. Firstly we consider this relation in the case of one-dimensional random walks. Afterwards we deal with higher-dimensional random walks. For comparison with the case of the quantum walk, we also treat the case of one-dimensional quantum walks. Our results bridge between the Ronkin function and the zeta function via quantum walks for the first time.Comment: 19 pages. arXiv admin note: substantial text overlap with arXiv:2202.05966; text overlap with arXiv:2109.07664, arXiv:2104.1028