Ong has shown that the modal mu-calculus model checking problem (equivalently, the alternating parity tree automaton (APT) acceptance problem) of possibly-infinite ranked trees generated by order-n recursion schemes is n-EXPTIME complete. We consider
two subclasses of APT and investigate the complexity of the respective acceptance problems. The main results are that, for APT with a single priority, the problem is still n-EXPTIME complete; whereas, for APT with a disjunctive transition function, the problem is (n-1)-EXPTIME complete. This study was motivated by Kobayashis recent work showing that the resource usage verification of functional programs can be reduced to the model checking of recursion schemes. As an application, we show that the resource usage verification problem is (n-1)-EXPTIME complete.
The statistics of isothermal lines and loops of the Cosmic Microwave Background (CMB) radiation on the sky map is studied and the fractal structure is confirmed in the radiation temperature fluctuation. We estimate the fractal exponents, such as the
fractal dimension $D_{mathrm{e}}$ of the entire pattern of isothermal lines, the fractal dimension $D_{mathrm{c}}$ of a single isothermal line, the exponent $zeta$ in Korv{c}aks law for the size distribution of isothermal loops, the two kind of Hurst exponents, $H_{mathrm{e}}$ for the profile of the CMB radiation temperature, and $H_{mathrm{c}}$ for a single isothermal line. We also perform fractal analysis of two artificial sky maps simulated by a standard model in physical cosmology, the WMAP best-fit $Lambda$ Cold Dark Matter ($Lambda$CDM) model, and by the Gaussian free model of rough surfaces. The temperature fluctuations of the real CMB radiation and in the simulation using the $Lambda$CDM model are non-Gaussian, in the sense that the displacement of isothermal lines and loops has an antipersistent property indicated by $H_{mathrm{e}} simeq 0.23 < 1/2$.
Fragment-size distributions have been studied experimentally in masticated viscoelastic food (fish sausage).The mastication experiment in seven subjects was examined. We classified the obtained results into two groups, namely, a single lognormal dist
ribution group and a lognormal distribution with exponential tail group. The facts suggest that the individual variability might affect the fragmentation pattern when the food sample has a much more complicated physical property. In particular, the latter result (lognormal distribution with exponential tail) indicates that the fragmentation pattern by human mastication for fish sausage is different from the fragmentation pattern for raw carrot shown in our previous study. The excellent data fitting by the lognormal distribution with exponential tail implies that the fragmentation process has a size-segregation-structure between large and small parts.In order to explain this structure, we propose a mastication model for fish sausage based on stochastic processes.
One-parameter family of discrete-time quantum-walk models on the square lattice, which includes the Grover-walk model as a special case, is analytically studied. Convergence in the long-time limit $t to infty$ of all joint moments of two components o
f walkers pseudovelocity, $X_t/t$ and $Y_t/t$, is proved and the probability density of limit distribution is derived. Dependence of the two-dimensional limit density function on the parameter of quantum coin and initial four-component qudit of quantum walker is determined. Symmetry of limit distribution on a plane and localization around the origin are completely controlled. Comparison with numerical results of direct computer-simulations is also shown.
We study a series of one-dimensional discrete-time quantum-walk models labeled by half integers $j=1/2, 1, 3/2, ...$, introduced by Miyazaki {it et al.}, each of which the walkers wave function has $2j+1$ components and hopping range at each time ste
p is $2j$. In long-time limit the density functions of pseudovelocity-distributions are generally given by superposition of appropriately scaled Konnos density function. Since Konnos density function has a finite open support and it diverges at the boundaries of support, limit distribution of pseudovelocities in the $(2j+1)$-component model can have $2j+1$ pikes, when $2j+1$ is even. When $j$ becomes very large, however, we found that these pikes vanish and a universal and monotone convex structure appears around the origin in limit distributions. We discuss a possible route from quantum walks to classical diffusion associated with the $j to infty$ limit.
A framework for discussing relationships between different types of games is proposed. Within the framework, quantum simultaneous games, finite quantum simultaneous games, quantum sequential games, and finite quantum sequential games are defined. In
addition, a notion of equivalence between two games is defined. Finally, the following three theorems are shown: (1) For any quantum simultaneous game G, there exists a quantum sequential game equivalent to G. (2) For any finite quantum simultaneous game G, there exists a finite quantum sequential game equivalent to G. (3) For any finite quantum sequential game G, there exists a finite quantum simultaneous game equivalent to G.
