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THE RICCI CURVATURE ON DIRECTED GRAPHS
Yamada, Taiki Korean Mathematical Society 2019 대한수학회지 Vol.56 No.1
In this paper, we consider the Ricci curvature of a directed graph, based on Lin-Lu-Yau's definition. We give some properties of the Ricci curvature, including conditions for a directed regular graph to be Ricci-flat. Moreover, we calculate the Ricci curvature of the cartesian product of directed graphs.
The Ricci curvature on directed graphs
Taiki Yamada 대한수학회 2019 대한수학회지 Vol.56 No.1
In this paper, we consider the Ricci curvature of a {\em directed} graph, based on Lin-Lu-Yau's definition. We give some properties of the Ricci curvature, including conditions for a directed regular graph to be Ricci-flat. Moreover, we calculate the Ricci curvature of the cartesian product of directed graphs.
Nakamae Toshio,Kamei Naosuke,Fujimoto Yoshinori,Yamada Kiyotaka,Tamura Takayuki,Tsuchikawa Yuji,Morisako Taiki,Harada Takahiro,Maruyama Toshiaki,Adachi Nobuo 대한척추외과학회 2022 Asian Spine Journal Vol.16 No.2
Study Design: Cross-sectional study. Purpose: We aimed to quantitatively assess bone marrow edema (BME) on magnetic resonance imaging (MRI) for patients with degenerative lumbar diseases. Overview of Literature: BME adjacent to a sclerotic endplate of the lumbar spine, detected using T2-weighted fat-saturated MRI, is closely associated with low back pain in patients with degenerative lumbar diseases. However, currently, there no quantitative evaluation methods for BME adjacent to the vertebral endplate. Methods: Patients with degenerative lumbar diseases, whose MRIs detected BME, were enrolled. On a T2-weighted fat-saturated MRI, BME appeared as a high-intensity region adjacent to the vertebral endplate. We calculated the contrast ratios (CRs) of BME and normal bone marrow using the signal intensities of BME, normal bone marrow, and the spinal cord. On computed tomography, we calculated Hounsfield unit (HU) values in the same area as BME, the sclerotic endplate, and normal bone marrow to assess bone density. Results: There were 16 men and 14 women, with an average age of 73.5 years. The mean CRs of BME and normal bone marrow were −0.015±0.056 and −0.407±0.023, respectively. BME’s CR was significantly higher than that of normal bone marrow (p<0.01). The HU values in the same area as BME, the sclerotic endplate, and normal bone marrow were 251.9±24.6, 828.3±35.6, and 98.1±9.3, respectively; these values were significantly different from each other (p<0.01). Conclusions: The CR on MRI is a useful quantitative assessment tool for BME in patients with degenerative lumbar diseases.