Abstract:
The exact renormalization group equation for pure quantum gravity is derived for an arbitrary gauge parameter in the space-time dimension $d=4$. This equation is given by a non-linear functional differential equation for the effective average action. An action functional of the effective average action is approximated by the same functional space of the Einstein-Hilbert action. From this approximation, $\beta$-functions for the dimensionless Newton constant and cosmological constant are derived non-perturbatively. These are used for an analysis of the phase structure and the ultraviolet non-Gaussian fixed point of the dimensionless Newton constant. This fixed point strongly depends on the gauge parameter and the cutoff function. However, this fixed point exists without these ambiguities, except for some gauges. Hence, it is possible that pure quantum gravity in $d=4$ is an asymptotically safe theory and non-perturbatively renormalizable.

Abstract:
The non-trivial ultraviolet fixed point in quantum gravity is calculated by means of the exact renormalization group equation in d-dimensions $(2\simeq d\leq4)$. It is shown that the ultraviolet non-Gaussian fixed point which is expected from the perturbatively $\epsilon$-expanded calculations in $2+\epsilon$ gravity theory remains in d=4. Hence it is possible that quantum gravity is an asymptotically safe theory and renormalizable in 2

Abstract:
We report empirical studies on the personal income distribution, and clarify that the distribution pattern of the lognormal with power law tail is the universal structure. We analyze the temporal change of Pareto index and Gibrat index to investigate the change of the inequality of the income distribution. In addition some mathematical models which are proposed to explain the power law distribution are reviewed.

Abstract:
We investigate the Japanese personal income distribution in the high income range over the 112 years 1887-1998, and that in the middle income range over the 44 years 1955-98. It is observed that the distribution pattern of the lognormal with power law tail is the universal structure. However the indexes specifying the distribution differ from year to year. One of the index characterizing the distribution is the mean value of the lognormal distribution; the mean income in the middle income range. It is found that this value correlates linearly with the Gross Domestic Product (GDP). To clarify the temporal change of the equality or inequality of the distribution, we analyze Pareto and Gibrat indexes, which characterize the distribution in the high income range and that in the middle income range respectively. It is found for some years that there is no correlation between the high income and the middle income. It is also shown that the mean value of Pareto index equals to 2, and the change of this index is effected by the change of the asset price. From these analysis we derive four constraints that must be satisfied by mathematical models.

Abstract:
Labor productivity was studied at the microscopic level in terms of distributions based on individual firm financial data from Japan and the US. A power-law distribution in terms of firms and sector productivity was found in both countries' data. The labor productivities were not equal for nation and sectors, in contrast to the prevailing view in the field of economics. It was found that the low productivity of the Japanese non-manufacturing sector reported in macro-economic studies was due to the low productivity of small firms.

Abstract:
We discuss the phase structure of the four-dimensional compact U(1) gauge theory at finite temperature using a deformation of the topological model. Its phase structure can be determined by the behavior of the Coulomb gas (CG) system on the cylinder. We utilize the relation between the CG system and the sine-Gordon (SG) model, and investigate the phase structure of the gauge theory in terms of the SG model. Especially, the critical-line equation of the gauge theory in the strong-coupling and high-temperature region is obtained by calculating the one-loop effective potential of the SG model.

Abstract:
Personal income distributions in Japan are analyzed empirically and a simple stochastic model of the income process is proposed. Based on empirical facts, we propose a minimal two-factor model. Our model of personal income consists of an asset accumulation process and a wage process. We show that these simple processes can successfully reproduce the empirical distribution of income. In particular, the model can reproduce the particular transition of the distribution shape from the middle part to the tail part. This model also allows us to derive the tail exponent of the distribution analytically.

Abstract:
In this paper, we investigate the structure and evolution of customer-supplier networks in Japan using a unique dataset that contains information on customer and supplier linkages for more than 500,000 incorporated non-financial firms for the five years from 2008 to 2012. We find, first, that the number of customer links is unequal across firms; the customer link distribution has a power-law tail with an exponent of unity (i.e., it follows Zipf's law). We interpret this as implying that competition among firms to acquire new customers yields winners with a large number of customers, as well as losers with fewer customers. We also show that the shortest path length for any pair of firms is, on average, 4.3 links. Second, we find that link switching is relatively rare. Our estimates indicate that the survival rate per year for customer links is 92 percent and for supplier links 93 percent. Third and finally, we find that firm growth rates tend to be more highly correlated the closer two firms are to each other in a customer-supplier network (i.e., the smaller is the shortest path length for the two firms). This suggests that a non-negligible portion of fluctuations in firm growth stems from the propagation of microeconomic shocks – shocks affecting only a particular firm – through customer-supplier chains.

Abstract:
Technological innovation has extensively been studied to make firms sustainable and more competitive. Within this context, the most important recent issue has been the dynamics of collaborative innovation among firms. We therefore investigated a patent network, especially focusing on its spatial characteristics. The results can be summarized as follows. (1) The degree distribution in a patent network follows a power law. A firm can then be connected to many firms via hubs connected to the firm. (2) The neighbors' average degree has a null correlation, but the clustering coefficient has a negative correlation. The latter means that there is a hierarchical structure and bridging different modules may shorten the paths between the nodes in them. (3) The distance of links not only indicates the regional accumulations of firms, but the importance of time it takes to travel, which plays a key role in creating links. (4) The ratio of internal links in cities indicates that we have to consider the existing links firms have to facilitate the creation of new links.

Abstract:
As complex networks in economics, we consider Japanese shareholding networks as they existed in 1985, 1990, 1995, 2000, 2002, and 2003. In this study, we use as data lists of shareholders for companies listed on the stock market or on the over-the-counter market. The lengths of the shareholder lists vary with the companies, and we use lists for the top 20 shareholders. We represent these shareholding networks as a directed graph by drawing arrows from shareholders to stock corporations. Consequently, the distribution of incoming edges has an upper bound, while that of outgoing edges has no bound. This representation shows that for all years the distributions of outgoing degrees can be well explained by the power law function with an exponential tail. The exponent depends on the year and the country, while the power law shape is maintained universally. We show that the exponent strongly correlates with the long-term shareholding rate and the cross-shareholding rate.