Space of spaces as a metric space

In spacetime physics, we frequently need to consider a set of all spaces (`universes') as a whole. In particular, the concept of `closeness' between spaces is essential. However, there has been no established mathematical theory so far which deals with a space of spaces in a suitable manner for spacetime physics. Based on the scheme of the spectral representation of geometry, we construct a space of all compact Riemannian manifolds equipped with the spectral measure of closeness. We show that this space of all spaces can be regarded as a metric space. We also show other desirable properties of this space, such as the partition of unity, locally-compactness and the second countability. These facts show that this space of all spaces can be a basic arena for spacetime physics.Comment: To appear in Communications in Mathematical Physics. 20 page

On the stationarity of Japanese-yen based purchasing power parity in the presence of the structural breaks

In this paper we perform the stationarity test on the Japanese-yen based real exchange rate of major trade partners of Japan, and we investigate the existences of relative PPPs with these countries and areas. The empirical analyses support that relative PPPs with the United States, Germany (with marks and euro), Korea and Taiwan are upheld when we consider the constant, linear trend and structural breaks. However, relative PPP is not upheld with China (Mainland).real exchange rate, relative PPP, stationarity, KPSS test, structural breaks

Numerical simulations of interacting disk galaxies

Galaxy-galaxy interactions have long attracted many extragalactic astronomers in various aspects. A number of computer simulations performed in the 1970s have successfully reproduced the peculiar morphologies observed in interacting disk galaxies and clarified that tidal deformation explains most of the observed global peculiarities. However, most of these simulations have used test particles in modelling the disk component. Tidal response of a self-gravitating disk remains to be further clarified. Another topic which is intensely discussed at present is the relation between galaxy-galaxy interactions and activity. Many observations suggest that interactions trigger strong starbursts and possibly active galactic nuclei (AGN). However, the detailed mechanism of triggering is not yet clear. It is vital here to understand the dynamics of interstellar gas. In order to understand various phenomena related to galaxy-galaxy interactions (mainly for disk galaxies), the author performed a series of numerical simulations on close galaxy encounters which includes both interstellar gas and self-gravitating disk components. In these simulations, the galaxy model to be perturbed (target galaxy) consists of a halo and a disk. The halo was treated as a rigid spherical gravitational field which is assumed to remain fixed during the interaction. The disk is composed of stars and gas. The stellar disk was constructed by 20000 collisionless particles of the same mass. Those particles move in the halo gravitational field, interacting with each other and with the perturber. Therefore, the self-graviy of the disk is properly taken into account. Stellar particles were initially given circular velocities with small random motions required to stabilize the disk against local axisymmetric disturbances. The gravitational field of the stellar disk was calculated by the particle-mesh scheme (e.g. Hockney and Eastwood 1981). The gaseous component was modelled by the cloud-particle scheme (e.g. Roberts and Hausman 1984). Here, the authors represent the gas as an ensemble of small spheres (i.e. clouds) and include the creation of an OB star in a cloud-cloud collision and subsequent velocity push on nearby clouds due to a supernova explosion

Holographic Renormalization Group Structure in Higher-Derivative Gravity

Classical higher-derivative gravity is investigated in the context of the holographic renormalization group (RG). We parametrize the Euclidean time such that one step of time evolution in (d+1)-dimensional bulk gravity can be directly interpreted as that of block spin transformation of the d-dimensional boundary field theory. This parametrization simplifies the analysis of the holographic RG structure in gravity systems, and conformal fixed points are always described by AdS geometry. We find that higher-derivative gravity generically induces extra degrees of freedom which acquire huge mass around stable fixed points and thus are coupled to highly irrelevant operators at the boundary. In the particular case of pure R^2-gravity, we show that some region of the coefficients of curvature-squared terms allows us to have two fixed points (one is multicritical) which are connected by a kink solution. We further extend our analysis to Minkowski time to investigate a model of expanding universe described by the action with curvature-squared terms and positive cosmological constant, and show that, in any dimensionality but four, one can have a classical solution which describes time evolution from a de Sitter geometry to another de Sitter geometry, along which the Hubble parameter changes drastically.Comment: 26 pages, 6 figures, typos correcte