This paper aims at an automatic evaluation of second language (L2) learners’ proficiencies and tries to analyze English conversation data having 94 statistics and Global Scale scores of the Common European Framework of Reference (CEFR) given to each participant. The CEFR defines Range, Accuracy, Fluency, Interaction and Coherence as 5 subcategories, which constitute the CEFR Global Scale score. The statistics were classified into the CEFR’s 5 subcategories. We used the Principal Component Analysis (PCA), an unsupervised machine learning method, on each subcategory and obtained the participants’ principal component scores (PC scores) of the 5 subcategories for estimation parameters. We predicted the participants’ CEFR Global scores using the Multiple Regression Analysis (MRA). The proposed prediction method using the PC scores was compared with conventional methods with the 94 statistics. Based on the coefficients of determination (R2), the value of the proposed method (0.82) was nearly equivalent to one of values obtained by the conventional methods. Meanwhile, as for standard deviation, the proposed method showed the smallest value in the comparison. The results indicated usability of the PCA and PC scores calculated from the CEFR subcategory data for objective evaluation of L2 learners’ English proficiencies.

A string figure is topologically a trivial knot lying on an imaginary plane orthogonal to the fingers with some crossings. The fingers prevent cancellation of these crossings. As a mathematical model of string figure we consider a knot diagram on the xy-plane in xyz-space missing some straight lines parallel to the z-axis. These straight lines correspond to fingers. We study minimal number of crossings of these knot diagrams under Reidemeister moves missing these lines.

A string figure is topologically a trivial knot lying on an imaginary plane orthogonal to the fingers with some crossings. The fingers prevent cancellation of these crossings. As a mathematical model of string figure, we consider a knot diagram on the [Formula: see text]-plane in [Formula: see text]-space missing some straight lines parallel to the [Formula: see text]-axis. These straight lines correspond to fingers. We study minimal number of crossings of these knot diagrams under Reidemeister moves missing these lines.