Bulletin of the Japan Society for Industrial and Applied Mathematics

Recently found unstable time-periodic solutions to the incompressible Navier-Stokes equation are reviewed to discuss their relevance to plane Couette turbulence and isotropic turbulence. It is shown that the periodic motion embedded in the Couette turbulence exhibits a regeneration cycle of near-wall coherent structures, which consists of formation and breakdown of streamwise vortices and low-velocity streaks. In phase space a turbulent state wanders around the corresponding periodic orbit for most of the time, so that the r.m.s. (root-mean-squares) of velocity fluctuations of the Couette turbulence agree very well with the temporal averages of those along the periodic orbit. The maximum values of the streamwise and wall-normal r.m.s. velocities for the periodic solution are compared with those for other known nonlinear solutions. The Kolmogorov universal-range energy spectrum is observed for the periodic motion embedded in high-symmetric turbulence at the Taylor-microscale Reynolds number Re_λ=67.

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Micro-phase separation of diblock copolymer melts have become an excellent model system for studying fundamental phenomena associated with molecular self-assembly. We deal with the gradient system derived from the free-energy functional of nonlocal type and focus on the morphological analysis of the periodic 3D structures obtained by the simulations. We have numerically confirmed that "balanced scaling law", i.e., a morphology is attained where two competing terms are balanced, is valid for the functional in the singular limit which is consistent with the theoretical results by Ren and Wei[19]. This view point allows us to demonstrate the mechanism behind the appearance of the gyroid minimizers in terms of some geometrical measures. We also apply computational homology to characterize the complex morphology during the phase transition dynamics. Our topological characterization points to the transient perforated lamellar state in the lamellar-hexagons transition and the t^<-1> law of the Betti number in the late stage of phaseordering process.

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Using Monte Carlo methods, we can simulate various phenomena in classical many-body systems. But Monte Carlo simulations for quantum systems are suffered from some particular problems for a long time. Recent developments of quantum Monte Carlo algorithms provide significant improvement in Monte Carlo simulations for some important quantum models. In this paper, such recent developments are reviewed from a unified perspective. In particular, two algorithms-loop and directed loop algorithms-are presented. Firstly, we will give the basis of quantum Monte Carlo method with path-integral representation. Quantum Monte Carlo algorithm gives importance sampling of configurations on path-integral representation, but a conventional one is not efficient. Secondly, we will introduce new novel variables in path-integral representation, which called "graph variable". And we will present loop and directed loop algorithms using such new graph variables.

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We analyze cross correlations between price fluctuations of different stocks, and obtain the eigenvalues λ of an equal-time crosscorrelation function C. The statistics of the eigenvalues of C are compared with those of the random matrix theory (RMT). We find that a majority of the eigenvalues of C fall within the RMT bounds [λ_-, λ_+]. We regard the range λ_+<λ as the principal part of correlations in the stock market. We construct portfolio of stocks by using the eigenvalues λ_+<λ, and report a performance of them.

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