Graph isomorphism completeness for trapezoid graphs

The complexity of the graph isomorphism problem for trapezoid graphs has been open over a decade. This paper shows that the problem is GI-complete. More precisely, we show that the graph isomorphism problem is GI-complete for comparability graphs of partially ordered sets with interval dimension 2 and height 3. In contrast, the problem is known to be solvable in polynomial time for comparability graphs of partially ordered sets with interval dimension at most 2 and height at most 2.Comment: 4 pages, 3 Postscript figure

Single-wavenumber Representation of Nonlinear Energy Spectrum in Elastic-Wave Turbulence of {F}\"oppl-von {K}\'arm\'an Equation: Energy Decomposition Analysis and Energy Budget

A single-wavenumber representation of nonlinear energy spectrum, i.e., stretching energy spectrum is found in elastic-wave turbulence governed by the F\"oppl-von K\'arm\'an (FvK) equation. The representation enables energy decomposition analysis in the wavenumber space, and analytical expressions of detailed energy budget in the nonlinear interactions are obtained for the first time in wave turbulence systems. We numerically solved the FvK equation and observed the following facts. Kinetic and bending energies are comparable with each other at large wavenumbers as the weak turbulence theory suggests. On the other hand, the stretching energy is larger than the bending energy at small wavenumbers, i.e., the nonlinearity is relatively strong. The strong correlation between a mode $a_{\bm{k}}$ and its companion mode $a_{-\bm{k}}$ is observed at the small wavenumbers. Energy transfer shows that the energy is input into the wave field through stretching-energy transfer at the small wavenumbers, and dissipated through the quartic part of kinetic-energy transfer at the large wavenumbers. A total-energy flux consistent with the energy conservation is calculated directly by using the analytical expression of the total-energy transfer, and the forward energy cascade is observed clearly.Comment: 11 pages, 4 figure

Weak and strong wave turbulence spectra for elastic thin plate

Variety of statistically steady energy spectra in elastic wave turbulence have been reported in numerical simulations, experiments, and theoretical studies. Focusing on the energy levels of the system, we have performed direct numerical simulations according to the F\"{o}ppl--von K\'{a}rm\'{a}n equation, and successfully reproduced the variability of the energy spectra by changing the magnitude of external force systematically. When the total energies in wave fields are small, the energy spectra are close to a statistically steady solution of the kinetic equation in the weak turbulence theory. On the other hand, in large-energy wave fields, another self-similar spectrum is found. Coexistence of the weakly nonlinear spectrum in large wavenumbers and the strongly nonlinear spectrum in small wavenumbers are also found in moderate energy wave fields.Comment: 5 pages, 3 figure

Large-scale lognormality in turbulence modeled by Ornstein-Uhlenbeck process

Lognormality was found experimentally for coarse-grained squared turbulence velocity and velocity increment when the coarsening scale is comparable to the correlation scale of the velocity (Mouri et al. Phys. Fluids 21, 065107, 2009). We investigate this large-scale lognormality by using a simple stochastic process with correlation, the Ornstein-Uhlenbeck (OU) process. It is shown that the OU process has a similar large-scale lognormality, which is studied numerically and analytically.Comment: 7 pages, 5 figures, PRE in pres