• Kyungpook Mathematical Journal
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• v.52 no.4
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• pp.513-519
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• 2012

For a spanning tree T of a connected graph ${\Gamma}$ and for a labelling ${\phi}$: E(T) ${\rightarrow}$ {+,-},${\phi}$ is called an alternating sign on a spanning tree T of a graph ${\Gamma}$ if for any cotree edge $e{\in}E({\Gamma})-E(T)$, the unique path in T joining both end vertices of e has alternating signs. In the present article, we prove that any graph has a spanning tree T and an alternating sign on T.

• Journal of The Korean Astronomical Society
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• v.46 no.3
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• pp.125-131
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• 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 Pharmacopuncture
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• v.18 no.1
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• pp.36-43
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• 2015

Objectives: Water-soluble carthami flos (WCF) is a new mixture of Carthami flos (CF) pharmacopuncture. We conducted a 4-week toxicity test of repeated intramuscular injections of WCF in Sprague-Dawley rats. Methods: Forty male and 40 female rats were divided into 4 groups of 10 male and 10 female SD rats: The control group received 0.5 mL/animal/day of normal saline whereas the three experimental groups received WCF at doses of 0.125, 0.25, and 0.5 mL/animal/day, respectively. For 4 weeks, the solutions were injected into the femoral muscle of the rats alternating from side to side. Clinical signs, body weights, and food consumption were observed; opthalmological examinations and urinalyses were performed. On day 29, blood samples were taken for hematological and clinical chemistry analyses. Then, necropsy was conducted in all animals to observe weights and external and histopathological changes in the bodily organs. All data were tested using a statistical analysis system (SAS). Results: No deaths were observed. Temporary irregular respiration was observed in male rats of the experimental group for the first 10 days. Body weights, food consumptions, opthalmological examinations, urinalyses, clinical chemistry analyses, organ weights and necropsy produced no findings with toxicological meaning. In the hematological analysis, delay of prothrombin time (PT) was observed in male rats of the 0.25- and the 0.5-mL/animal/day groups. In the histopathological test, a dose-dependent inflammatory cell infiltration into the fascia and panniculitis in perimuscular tissues was observed in all animals of the experimental groups. However, those symptoms were limited to local injection points. No toxicological meanings, except localized changes, were noted. Conclusion: WCF solution has no significant toxicological meaning, but does produce localized symptoms. No observed adverse effect level (NOAEL) of WCF in male and female rats is expected for doses over 0.5 mL/animal/day.