• Communications of the Korean Mathematical Society
• /
• v.12 no.1
• /
• pp.37-43
• /
• 1997

Some formulas of the genus numbers and the ambiguous ideal class numbers of function fields are given and these numbers are shown to be the same when the extension is cyclic.

• 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.

• Journal of the Korean Mathematical Society
• /
• v.53 no.6
• /
• pp.1431-1443
• /
• 2016

Let k be a global field of characteristic unequal to two. Let $C:y^2=f(x)$ be a nonsingular projective curve over k, where f(x) is a quartic polynomial over k with nonzero discriminant, and K = k(C) be the function field of C. For each prime spot p on k, let ${\hat{k}}_p$ denote the corresponding completion of k and ${\hat{k}}_p(C)$ the function field of $C{\times}_k{\hat{k}}_p$. Consider the map $$h:Br(K){\rightarrow}{\prod\limits_{\mathfrak{p}}}Br({\hat{k}}_p(C))$$, where p ranges over all the prime spots of k. In this paper, we explicitly describe all the constant classes (coming from Br(k)) lying in the kernel of the map h, which is an obstruction to the Hasse principle for the Brauer groups of the curve. The kernel of h can be expressed in terms of quaternion algebras with their prime spots. We also provide specific examples over ${\mathbb{Q}}$, the rationals, for this kernel.

• Korean Journal of Ecology and Environment
• /
• v.51 no.1
• /
• pp.40-59
• /
• 2018

This review summarizes the history of freshwater zooplankton research in Korean Society of Limnology and necessary future topics that are remain poorly investigated in South Korea based on main research topics of published articles focusing on lakes, reservoirs, rivers and wetland ecosystems. In Korea, a total 450 freshwater zooplankton species have been reported (85 species of cladocera, ca. 230 species of rotifera since 1939, and 138 species of copepoda), and they cover 10% of total zooplankton species list. In the present paper, we provide recent species list of zooplankton found in Korea and their taxonomic keys. Over periods of 45 years, there are 25 published papers for zooplankton biota in lentic ecosystems in Korean Journal of Ecology and Environment (KJEE). The ecology of zooplankton communities in rivers has focused on the mechanisms involved in regulating their abundance, diversity and spatio-temporal patterns of genus Bosmina and rotifers (genus Brachionus and Keratella) that are most frequently found from Korea. On the other hand, the studies on zooplankton in wetland has focused on Alona, Chydorus and Ceriodaphnia with special emphasis on their relationships with aquatic macrophytes. Even though studies on the freshwater ecosystem in Korea have a long history, a few of studies on zooplankton biota were conducted at rice paddy, Dumbeong and wetlands. During the last two decades, experimental advances in freshwater zooplankton ecology and understanding of structure and function of this biota were made from a series of articles mainly in journal of KJEE. For future studies, quantitative, experimental and interdisciplinary approaches would be key words to understand zooplankton ecology and their roles in aquatic ecosystems under changing environments, and we have suggested necessary zooplankton research fields and future directions.