Eigenstate Transition of Multi-Channel Time Series Data around Earthquakes

To decrease human and economic damage owing to earthquakes, it is necessary to discover signals preceding earthquakes. We focus on the concept of “early warning signals” developed in bifurcation analysis, in which an increase in the variances of variables precedes its transition. If we can treat earthquakes as one of the transition phenomena that moves from one state to the other state, this concept is useful for detecting earthquakes before they start. We develop a covariance matrix from multi-channel time series data observed by an observatory on the seafloor and calculate the first eigenvalue and corresponding eigenstate of the matrix. By comparing the time dependence of the eigenstate to some past earthquakes, it is shown that the contribution from specific observational channels to the eigenstate increases before earthquakes, and there is a case in which the eigenvalue increases as predicted in early warning signals. This result suggests the first eigenvalue and eigenstate of multi-channel data are useful to identify signals preceding earthquakes.

The dual eigenvalue problems of the conformable fractional Sturm–Liouville problems

In this paper, we are concerned with the eigenvalue gap and eigenvalue ratio of the Dirichlet conformable fractional Sturm–Liouville problems. We show that this kind of differential equation satisfies the Sturm–Liouville property by the Prüfer substitution. That is, the nth eigenfunction has $n-1$ n − 1 zero in $( 0,\pi ) $ ( 0 , π ) for $n\in \mathbb{N}$ n ∈ N . Then, using the homotopy argument, we find the minimum of the first eigenvalue gap under the class of single-well potential functions and the first eigenvalue ratio under the class of single-barrier density functions. The result of the eigenvalue gap is different from the classical Sturm–Liouville problem.