• Korean Institute of Interior Design Journal
• /
• v.14 no.5 s.52
• /
• pp.26-34
• /
• 2005

The systematic way of the boundary thought in Buddhism, when applied to the principles of building, determines certain forms to certain temples, and organizes their topological boundary concept structure - the continuous experience of the visitor from his/her entry bridge(connecting), through the main temple gate(neighbourhood), pavilion gate(including), stairs(continuance), to the arrival at the pavilion of the god of a mountain(spiral), which reconstitutes the Buddhist boundary symbolism and philosophy. The topological boundary spaces of temples are an architectural manifestation of Buddhism's Mahayana boundary concept aspects, whose object is to play a productive and active role in the enlightenment of people, serving the very basic end of the religion. The disciplined topological boundary spaces of the temple, as a reification of the boundary symbolisms of Buddhist topological cosmology, corresponds to Buddha-Ksetra, the highest state of existence in the universe. Visitors to the temple are invited to participate in the world of abundant Buddhist boundary concept symbols, and through this process, is enabled to elevate oneself to the transcendent topological boundary world and have a simulated experience of liberation.

• Communications of the Korean Mathematical Society
• /
• v.37 no.1
• /
• pp.279-292
• /
• 2022

In this paper we adhere to the definition of infra-topological space as it was introduced by Al-Odhari. Namely, we speak about families of subsets which contain ∅ and the whole universe X, being at the same time closed under finite intersections (but not necessarily under arbitrary or even finite unions). This slight modification allows us to distinguish between new classes of subsets (infra-open, ps-infra-open and i-genuine). Analogous notions are discussed in the language of closures. The class of minimal infra-open sets is studied too, as well as the idea of generalized infra-spaces. Finally, we obtain characterization of infra-spaces in terms of modal logic, using some of the notions introduced above.

• The Bulletin of The Korean Astronomical Society
• /
• v.43 no.2
• /
• pp.41.2-41.2
• /
• 2018

The Minkowski Functionals of the matter densityeld, as traced by galaxies, contain information regarding the nature of dark energy and the fraction of dark matter in the Universe. In particular, the genus is a statistic that provides a clean measurement of the shape of the linear matter power spectrum. As the genus is a topological quantity, it is insensitive to galaxy bias and gravitational collapse. Furthermore, as it traces the linear matter power spectrum, it is a conserved quantity with redshift. Hence the genus amplitude is a standard population that can be used to test the distance-redshift relation. In this talk, I show how we can extract the genus from galaxy catalogs, and how we can use its properties to constrain the equation of state of dark energy and the energy content of the Universe.

• The Bulletin of The Korean Astronomical Society
• /
• v.35 no.2
• /
• pp.42.2-42.2
• /
• 2010

The possibility of detection of deviation from Gaussian distribution of primordial perturbations in the Cosmic Microwave Background (CMB) Radiation is very important because it can shed light on how the perturbations were created in the very early universe. We study the effect of the primordal non-Gaussianity on topological observables called Minkowski Functionals, which are functions of the temperature fluctuation field, and show that they carry distinct signatures of different types of non-Gaussianities. Then, we constrain the non-Gaussianity parameters by comparing the theoretical predictions of the Minkowski Functionals with measurements from observational data from WMAP.

• The Bulletin of The Korean Astronomical Society
• /
• v.38 no.2
• /
• pp.56.2-56.2
• /
• 2013

We study the three-dimensional genus topology of large-scale structure using the CMASS Data Release 11 sample of the SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS). The CMASS sample yields a genus curve that is characteristic of one produced by Gaussian random-phase initial conditions. The data thus supports the standard model of inflation where random quantum fluctuations in the early universe produced Gaussian random-phase initial conditions. Modest deviations in the observed genus from random phase are as expected from the nonlinear evolution of structure. We construct mock SDSS CMASS surveys along the past light cone from the Horizon Run 3 (HR3) N-body simulations, where gravitationally bound dark matter subhalos are identified as the sites of galaxy formation. We study the genus topology of the HR3 mock surveys with the same geometry and sampling density as the observational sample, and the observed genus topology to be consistent with LCDM as simulated by the HR3 mock samples.

• The Bulletin of The Korean Astronomical Society
• /
• v.43 no.2
• /
• pp.36.3-37
• /
• 2018

By utilizing large-scale graph analytic tools in the modern Big Data platform, Apache Spark, we investigate the topological structures of five different multiverses produced by cosmological n-body simulations with various cosmological initial conditions: (1) one standard universe, (2) two different dark energy states, and (3) two different dark matter densities. For the Big Data calculations, we use a custom build of stand-alone Spark cluster at KIAS and Dataproc Compute Engine in Google Cloud Platform with the sample sizes ranging from 7 millions to 200 millions. Among many graph statistics, we find that three simple graph measurements, denoted by (1) $n_\k$, (2) $\tau_\Delta$, and (3) $n_{S\ge5}$, can efficiently discern different topology in discrete point distributions. We denote this set of three graph diagnostics by kT5+. These kT5+ statistics provide a quick look of various orders of n-points correlation functions in a computationally cheap way: (1) $n = 2$ by $n_k$, (2) $n = 3$ by $\tau_\Delta$, and (3) $n \ge 5$ by $n_{S\ge5}$.

• Journal of The Korean Astronomical Society
• /
• v.29 no.spc1
• /
• pp.19-21
• /
• 1996

We apply topological measures of clustering such as percolation and genus curves (PC & GC) and shape statistics to a set of scale free N-body simulations of large scale structure. Both genus and percolation curves evolve with time reflecting growth of non-Gaussianity in the N-body density field. The amplitude of the genus curve decreases with epoch due to non-linear mode coupling, the decrease being more noticeable for spectra with small scale power. Plotted against the filling factor GC shows very little evolution - a surprising result, since the percolation curve shows significant evolution for the same data. Our results indicate that both PC and GC could be used to discriminate between rival models of structure formation and the analysis of CMB maps. Using shape sensitive statistics we find that there is a strong tendency for objects in our simulations to be filament-like, the degree of filamentarity increasing with epoch.

• Journal of The Korean Astronomical Society
• /
• v.46 no.3
• /
• pp.125-131
• /
• 2013

We present the relation between the genus in cosmology and the Betti numbers for excursion sets of three- and two-dimensional smooth Gaussian random fields, and numerically investigate the Betti numbers as a function of threshold level. Betti numbers are topological invariants of figures that can be used to distinguish topological spaces. In the case of the excursion sets of a three-dimensional field there are three possibly non-zero Betti numbers; ${\beta}_0$ is the number of connected regions, ${\beta}_1$ is the number of circular holes (i.e., complement of solid tori), and ${\beta}_2$ is the number of three-dimensional voids (i.e., complement of three-dimensional excursion regions). Their sum with alternating signs is the genus of the surface of excursion regions. It is found that each Betti number has a dominant contribution to the genus in a specific threshold range. ${\beta}_0$ dominates the high-threshold part of the genus curve measuring the abundance of high density regions (clusters). ${\beta}_1$ dominates the genus near the median thresholds which measures the topology of negatively curved iso-density surfaces, and ${\beta}_2$ corresponds to the low-threshold part measuring the void abundance. We average the Betti number curves (the Betti numbers as a function of the threshold level) over many realizations of Gaussian fields and find that both the amplitude and shape of the Betti number curves depend on the slope of the power spectrum n in such a way that their shape becomes broader and their amplitude drops less steeply than the genus as n decreases. This behaviour contrasts with the fact that the shape of the genus curve is fixed for all Gaussian fields regardless of the power spectrum. Even though the Gaussian Betti number curves should be calculated for each given power spectrum, we propose to use the Betti numbers for better specification of the topology of large scale structures in the universe.