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BETTI NUMBERS OF GAUSSIAN FIELDS
Park, Changbom,Pranav, Pratyush,Chingangbam, Pravabati,Van De Weygaert, Rien,Jones, Bernard,Vegter, Gert,Kim, Inkang,Hidding, Johan,Hellwing, Wojciech A. The Korean Astronomical Society 2013 Journal of The Korean Astronomical Society Vol.46 No.3
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.
Novel mutations of CDKN1C in Japanese patients with Beckwith-Wiedemann syndrome
Hitomi Yatsuki,Ken Higashimoto,Kosuke Jozaki,Kayoko Koide,Junichiro Okada,Yoriko Watanabe,Nobuhiko Okamoto,Yoshinobu Tsuno,Yoko Yoshida,Kazutoshi Ueda,Kenji Shimizu,Hirofumi Ohashi,Tsunehiro Mukai,Hid 한국유전학회 2013 Genes & Genomics Vol.35 No.2
Beckwith-Wiedemann syndrome (BWS) is an imprinting-related human disease that is characterized by macrosomia, macroglossia, abdominal wall defects, and variable minor features. BWS is caused by several genetic/epigenetic alterations, such as loss of methylation at KvDMR1,gain of methylation at H19-DMR, paternal uniparental disomy of chromosome 11, CDKN1C mutations, and structural abnormalities of chromosome 11. CDKN1C is an imprinted gene with maternal preferential expression, encoding for a cyclin-dependent kinase (CDK) inhibitor. Mutations in CDKN1C are found in 40 % of familial BWS cases with dominant maternal transmission and in *5 % of sporadic cases. In this study, we searched for CDKN1C mutations in 37BWS cases that had no evidence for other alterations. We found five mutations—four novel and one known—from a total of six patients. Four were maternally inherited and one was a de novo mutation. Two frame-shift mutations and one nonsense mutation abolished the QT domain, containing a PCNA-binding domain and a nuclear localization signal. Two missense mutations occurred in the CDK inhibitory domain,diminishing its inhibitory function. The above-mentioned mutations were predicted by in silico analysis to lead to loss of function; therefore, we strongly suspect that such anomalies are causative in the etiology of BWS.