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The objective of this paper is to formulate a statistical method of testing the hypothesis that the distribution of activity points (such as retail stores) is independent of location of 'surface-like' infrastructural elements (such as parks). In order to do this, first, the probability density function of a distance from a random point to the nearest surface-like element is derived. Second, through the use of this function, a measure, R, of spatial dependency on the surface-like elements is defined as the ratio of the average nearest-neighbor distance to the expected average nearest-neighbor distance. This measure is an extension of the ordinary nearest-neighbor distance measure frequently referred to in geography and ecology. Third, the statistical use of measure R is shown. Fourth, as this measure is difficult to compute geometrically, the computational method of calculating the value of R is developed. Last, by use of this method, a test is conducted to decide whether or not the distribution of high-class apartment buildings in Setagaya, Tokyo, is affected by the location of big parks.

Context. Nuclear star clusters (NSCs) at the dynamical center of galaxies appear to have a complex star formation history. This suggests repeated star formation, even in the influence of the strong tidal field from supermassive black holes. Although the central region of our Galaxy is an ideal target for studies of the star formation history in the NSCs, most studies in the past have concentrated on a projected distance of R = 0.5-3 pc from the supermassive black hole Sgr A∗. Aims. In our previous study, we detected 31 so far unknown early-type star candidates throughout the Galactic NSC (at R = 0.5-3 pc). They were found via near-infrared (NIR) imaging observations with narrow-band filters which are sensitive to CO absorption lines at ~2.3 μm, a prominent feature for old, late-type stars. The aim of this study is to confirm the spectral type for the early-type star candidates. Methods. We have carried out NIR spectroscopic observations of the early-type star candidates using Subaru/IRCS/AO188 and the laser guide star system. K-band spectra for 20 out of the 31 candidates and reference late-type stars were obtained. By determining an equivalent width, EW(CO), of the CO absorption feature at ≠2.294 μm, we have derived an effective temperature and a bolometric magnitude for each candidate and late-type star, and then constructed an HR diagram. Results. No young (~Myr) massive stars are included in the 20 candidates we observed; however, 13 candidates are most likely intermediate-age giants (50-500 Myr). Two other sources have ages of ~1 Gyr and the remaining five sources are old (>1 Gyr), late-type giants. Conclusions. Although none of the early-type star candidates from our previous narrow-band imaging observations can be confirmed as a young star, we find that the photometric technique can distinguish old, late-type giants from young and intermediate-age populations. From the 20 spectroscopically observed candidates, 65% of them are confirmed as being younger than 500 Myr. The intermediate-age stars could be as yet unknown members of a population formed in a starburst ~100 Myr ago. Finding no young (~a few Myr) stars at R = 0.5-3 pc favors the in situ formation scenario for the presence of the young stars at R< 0.5 pc, although we do not completely exclude the possible existence of unknown young, massive stars in the region from our observations. Furthermore, the different spatial distributions of the young and the intermediate-age stars imply that the Galactic NSC is an aggregate of stars that were born in different places and under a variety of physical conditions.© 2016 ESO. ; This work was supported by KAKENHI, Grant-in-Aid for Research Activity Start-up 23840044, Specially Promoted Research 22000005, COE Research 23103001 and 24103508, Grant-in-Aid for Exploratory Research 15K13463, Grant-in-Aid for challenging Exploratory Research 15K13463, and Young Scientists (A) 25707012, and Institutional Program for Young Researcher Overseas Visits. R.S. acknowledges support by grants AYA2010-17631 and AYA2009-13036 of the Spanish Ministry of Economy and Competition, and by grant P08-TIC-4075 of the Junta de Andalucia. R.S. acknowledges support by the Ramon y Cajal programme of the Spanish-Ministry of Economy and Competition. This material is partly based upon work supported, in part, by the National Science Foundation Grant No. 1066293 and the hospitality of the Aspen Center for Physics. The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement No. [614922]. ; Peer Reviewed