Considering relativistic symmetry as the first principle of quantum mechanics

On the basis of the relativistic symmetry of Minkowski space, we derive a Lorentz invariant equation for a spread electron. This equation slightly differs from the Dirac equation and includes additional terms originating from the spread of an electron. Further, we calculate the anomalous magnetic moment based on these terms. These calculations do not include any divergence; therefore, renormalization procedures are unnecessary. In addition, the relativistic symmetry existing among coordinate systems will provide a new prospect for the foundations of quantum mechanics like the measurement process.Comment: LaTeX2e, 18 pages, no figure; The subsection Self-energy estimation was improved; Accepted for publication in EJT

Trial calculation of relating the equilibrium state of minerals to the descriptive mineralogy

Statistical calculation have been carried out on the volumes of the asymmetric unit of minerals. This treatment is related to the origin of the symmetry and periodicity of the crystals, and to the equilibrium conditions of these crystals. From the view point of the cohesion energy, if the crystals were grown under the condition of nearly perfect equilibrium states, than all the volumes of the asymmetric unit of each crystal structure will be approximately equal, and if the volume of the asymmetric units of a certain mineral is larger than the average value, this mineral is considered to be grown in a metastable condition. The calculation of the cell dimensions of minerals have been carried out by the use of deta from previous investigations. The statistical consideration of the volumes of the asymmetric unit of minerals is considered to be an appropriate criterion to relate the stability of minerals to their descriptive mineralogy

Global small amplitude solutions for two-dimensional nonlinear Klein-Gordon systems in the presence of mass resonance

We consider a nonlinear system of two-dimensional Klein-Gordon equations with masses satisfying the resonance relation. We introduce a structural condition on the nonlinearities under which the solution exists globally in time and decays at the rate $O(|t|^{-1})$. In particular, our new condition includes the Yukawa type interaction, which has been excluded from the null condition in the sense of J.-M.Delort, D.Fang and R.Xue.Comment: to appear in J. Differential Equation

Dynamic mode decomposition in vector-valued reproducing kernel Hilbert spaces for extracting dynamical structure among observables

Understanding nonlinear dynamical systems (NLDSs) is challenging in a variety of engineering and scientific fields. Dynamic mode decomposition (DMD), which is a numerical algorithm for the spectral analysis of Koopman operators, has been attracting attention as a way of obtaining global modal descriptions of NLDSs without requiring explicit prior knowledge. However, since existing DMD algorithms are in principle formulated based on the concatenation of scalar observables, it is not directly applicable to data with dependent structures among observables, which take, for example, the form of a sequence of graphs. In this paper, we formulate Koopman spectral analysis for NLDSs with structures among observables and propose an estimation algorithm for this problem. This method can extract and visualize the underlying low-dimensional global dynamics of NLDSs with structures among observables from data, which can be useful in understanding the underlying dynamics of such NLDSs. To this end, we first formulate the problem of estimating spectra of the Koopman operator defined in vector-valued reproducing kernel Hilbert spaces, and then develop an estimation procedure for this problem by reformulating tensor-based DMD. As a special case of our method, we propose the method named as Graph DMD, which is a numerical algorithm for Koopman spectral analysis of graph dynamical systems, using a sequence of adjacency matrices. We investigate the empirical performance of our method by using synthetic and real-world data.Comment: 34 pages with 4 figures, Published in Neural Networks, 201