Recognition of plane-to-plane map-germs

We present a complete set of criteria for determining A-types of plane-to-plane map-germs of corank one with A-codimension <7, which provides a new insight into the A-classification theory from the viewpoint of recognition problem. As an application to generic differential geometry, we discuss about projections of smooth surfaces in 3-space.Comment: 22 page

Binary differential equations at parabolic and umbilical points for 22-parameter families of surfaces

We determine local topological types of binary differential equations of asymptotic curves at parabolic and flat umbilical points for generic $2$-parameter families of surfaces in $\mathbb P^3$ by comparing our projective classification of Monge forms and classification of general BDE obtained by Tari and Oliver. In particular, generic bifurcations of the parabolic curve are classified. The flecnodal curve is also examined by direct computations, and we present new bifurcation diagrams in typical examples.Comment: 20 page

A novel symmetry in nanocarbons: pre-constant discrete principal curvature structure

Since the first-principles calculations in quantum chemistry precisely provide possible configurations of carbon atoms in nanocarbons, we have analyzed the geometrical structure of the possible carbon configurations and found that there exists a novel symmetry in the nanocarbons, i.e., the pre-constant discrete principal curvature (pCDPC) structure. In terms of the discrete principal curvature based on the discrete geometry for trivalent oriented graphs developed by Kotani, Naito, and Omori (Comput. Aided Geom. Design, $\bf{58}$, (2017), 24-54), we numerically investigated discrete principal curvature distribution of the nanocarbons, C$_{60}$, carbon nanotubes, C$_{120}$ (C$_{60}$ dimer), and C$_{60}$-polymers (peanut-shaped fullerene polymers). While the C$_{60}$ and nanotubes have the constant discrete principal curvature (CDPC) as we expected, it is interesting to note that the C$_{60}$-polymers and C$_{60}$ dimer also have the almost constant discrete principal curvature, i.e., pCDPC, which is surprising. A nontrivial pCDPC structure with revolutionary symmetry is available due to discreteness, though it has been overlooked in geometry. In discrete geometry, there appears a center axisoid which is the discrete analogue of the center axis in the continuum differential geometry but has three-dimensional structure rather than a one-dimensional curve due to its discrete nature. We demonstrated that such pCDPC structure exists in nature, namely in the C$_{60}$-polymers. Furthermore, since we found that there is a positive correlation between the degree of the CDPC structure and stability of the configurations for certain class of the C$_{60}$-polymers, we also revealed the origin of the pCDPC structure from an aspect of materials science.Comment: 18 page

Three-dimensional topological radiogenomics of epidermal growth factor receptor Del19 and L858R mutation subtypes on computed tomography images of lung cancer patients

Objectives : To elucidate a novel radiogenomics approach using three-dimensional (3D) topologically invariant Betti numbers (BNs) for topological characterization of epidermal growth factor receptor (EGFR) Del19 and L858R mutation subtypes. Methods : In total, 154 patients (wild-type EGFR, 72 patients; Del19 mutation, 45 patients; and L858R mutation, 37 patients) were retrospectively enrolled and randomly divided into 92 training and 62 test cases. Two support vector machine (SVM) models to distinguish between wild-type and mutant EGFR (mutation [M] classification) as well as between the Del19 and L858R subtypes (subtype [S] classification) were trained using 3DBN features. These features were computed from 3DBN maps by using histogram and texture analyses. The 3DBN maps were generated using computed tomography (CT) images based on the Čech complex constructed on sets of points in the images. These points were defined by coordinates of voxels with CT values higher than several threshold values. The M classification model was built using image features and demographic parameters of sex and smoking status. The SVM models were evaluated by determining their classification accuracies. The feasibility of the 3DBN model was compared with those of conventional radiomic models based on pseudo-3D BN (p3DBN), two-dimensional BN (2DBN), and CT and wavelet-decomposition (WD) images. The validation of the model was repeated with 100 times random sampling. Results : The mean test accuracies for M classification with 3DBN, p3DBN, 2DBN, CT, and WD images were 0.810, 0.733, 0.838, 0.782, and 0.799, respectively. The mean test accuracies for S classification with 3DBN, p3DBN, 2DBN, CT, and WD images were 0.773, 0.694, 0.657, 0.581, and 0.696, respectively. Conclusion : 3DBN features, which showed a radiogenomic association with the characteristics of the EGFR Del19/L858R mutation subtypes, yielded higher accuracy for subtype classifications in comparison with conventional features